CUET Mathematics Calculus > Differential Equations PYQs

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

Match List-I with List-II | List-I | List-II | |------------|-------------| | (A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$ | (I) 1 | | (B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$ | (II) 4 | | (C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$ | (III) not defined | | (D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$ | (IV) 2 | Choose the correct answer from the options given below:

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$ (A) Integrating factor is $2x$ (B) Integrating factor is $x^2$ (C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant. (D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant. Choose the correct answer from the options given below:

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

The sum of the order and degree of the differential equation representing the family of curves $y = mx + m^4$, where m is arbitrary constant, is

The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$, given that $y = 1$ when $x = 1$ is:

If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$, then the value of $m + n$ is:

Consider the differential equation $xdy = (y + 2x^3)dx$. Then which of the following are TRUE? (A) It is a homogeneous differential equation. (B) Product of the order and degree of the differential equation in one. (C) Integrating factor is x. (D) General solution of the differential equation is $y = x^3 + Cx$, where C is an arbitary constant. Choose the *correct* answer from the options given below:

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Sum of order and degree** | | (A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$ | (I) 2 | | (B) $\frac{dy}{dx} = \sin(x + y)$ | (II) 3 | | (C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$ | (III) 4 | | (D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$ | (IV) 5 | Choose the **correct** answer from the options given below:

Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)

The solution of the differential equation $ydx + (x - y^2)dy = 0$ is

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of differential equation $\frac{d^3y}{dx^3} = e^{\frac{dx}{dy}}$ | (I) 2 | | (B) The order of differential equation $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$ | (II) 4 | | (C) The sum of order and degree of differential equation $\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^5 = x$ | (III) not defined | | (D) The number of arbitrary constants in the general solution of a differential equation of order 2 | (IV) 3 | Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

The solution of the differential equation $\frac{dy}{dx} = \sqrt\frac{{y}}{x}$ is

The order of $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \left[a \frac{d^2y}{dx^2}\right]^{\frac{1}{2}}$ is

The general solution of the differential equation $x\left(\frac{dy}{dx}\right) = y + x \tan\left(\frac{y}{x}\right)$ is

The sum of order and degree of the differential equation $y = x\frac{dy}{dx} + 2\sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ is

Which of the following are first order linear differential equations? (A) $\frac{dx}{dy} + P_1(y)x = Q_1(y)$ : $P_1(y)$ and $Q_1(y)$ are functions of y or constant functions (B) $\frac{dy}{dx} + P_2(x)y = Q_2(x)$ : $P_2(x)$ and $Q_2(x)$ are functions of x or constant functions (C) $(x + y)\frac{dy}{dx} = x - 2y$ (D) $(1 + x^2)\frac{dy}{dx} - 2xy = x^2 + 3$ Choose the correct answer from the options given below:

The solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 5x + 2y$ is given by

For the differential equation $ydx - (x + 3y^2)dy = 0$, which of the following statements are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $x = 3y^2 + Cy$ : $C$ is an arbitrary constant (D) If $y(0) = 1$, then its particular solution is $x = 3y^2 - 1$ Choose the correct answer from the options given below:

The product of order and degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^2 + x^2y\left(\frac{d^2y}{dx^2}\right)^3 = 2x^5$ is:

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$, $x \neq 0$ where A and B are arbitrary constants, is given by

Match List-I with List-II: | List-I | List-II | | --- | --- | | **Differential Equations** | **Degree/Order** | | (A) Degree of the differential equation $\frac{d^3y}{dx^3} + 2 \log x.y = 0$ | (I) 3 | | (B) Order of the differential equation $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^4 + xy = 0$ | (II) 2 | | (C) Degree of the differential equation $\left(\frac{d^4y}{dx^4}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^2y = 0$ | (III) 1 | | (D) Order of the differential equation $\frac{d^3y}{dx^3} + y\left(\frac{dy}{dx}\right)^3 = 0$ | (IV) 4 |

The particular solution of the differential equation $\frac{dy}{dx} = 8yx$ when $y = 1$ at $x = 0$

For the differential equation $x\frac{dy}{dx} + 3y = x^2\log_e x$, which of the following statements are TRUE? (A) Product of order and degree is 1 (B) Integrating factor is $x^3$ (C) Integrating factor is $3x$ (D) General solution is $y = \frac{x^3}{36}(6\log_e|x| - 1) + Cx^{-3}$, C is an arbitrary constant. Choose the correct answer from the options given below:

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

The particular solution of the differential equation $\frac{dy}{dx} + \frac{3y}{x} = 0$, $y(1) = 1$ is

Which one of the following equations is a homogeneous differential equation?

The solution of the differential equation $(x^2 + xy)dy = (x^2 + y^2)dx$ is

Match List-I with List-II | List-I | List-II | |---|---| | (A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$ | (I) 4 | | (B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$ | (II) 1 | | (C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$ | (III) Not defined | | (D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$ | (IV) 3 | Choose the correct answer from the options given below:

In the following differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ order and degree is:

The solution of the differential equation $xdy - ydx = 0$ represents

The particular solution of the differential equation $\frac{dy}{dx} = e^{x^2/2} + xy$, when $x = 0$, $y = 1$, is

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equations | Order and degree | | (A) $ydx + x\log(y/x)dy - 2xdy = 0$ | (I) Order : 2, degree:1 | | (B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$ | (II) Order :1, degree:1 | | (C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$ | (III) Order : 3, degree:2 | | (D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$ | (IV) Order : 1, degree: Not defined | Choose the correct answer from the options given below:

Curd is at 80° F, five minutes later it came down at 60°F. After another 5 minutes, its temperature became 50° F. Given that the rate of change of temperature is proportional to (T - S), where S is temperature of the surroundings and T is temperature of the curd at any time t. Then the temperature of the surroundings is :

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Order and Degree** | | (A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$ | (I) order = 1 and degree = 2 | | (B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$ | (II) order = 2 and degree = 1 | | (C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$ | (III) order = 2 and degree = 2 | | (D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$ | (IV) order = 2 and degree = 4 | Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$, $y = \frac{\pi}{4}$ when $x = 1$ is

Match List-I with List-II | List-I | List-II | |---|---| | **Differential Equation** | **Integrating Factor** | | (A) $\frac{dy}{dx} + 2xy = 1$ | (I) $x$ | | (B) $x\frac{dy}{dx} + 2xy = 1$ | (II) $e^{2x}$ | | (C) $x\frac{dy}{dx} + y = 1$ | (III) $x^2$ | | (D) $x\frac{dy}{dx} + 2y = 2$ | (IV) $e^{x^2}$ | Choose the correct answer from the options given below:

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equation** | **Degree** | | (A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$ | (I) 3 | | (B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$ | (II) 1 | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$ | (III) not defined | | (D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$ | (IV) 2 | Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dr}{dt} = -rt, r(0) = r_0$ is

The solution of the differential equation $\frac{dy}{dx} - \frac{y}{x} = 2\log_e x$

Which of the following statements is/are true? (A) $(\tan^{-1} y - x)dy = (1 + y^2)dx$ is a differential equation where variables are separable. (B) $(1 + x^2)dy + 2xydy = \cot x \ dx (x \neq 0)$ is a first order linear differential equation. (C) $(4x + 6y + 5)dy - (3y + 2x + 4)dx = 0$ is not a homogeneous differential equation. (D) $(xy)dx - (x + y^2)dy = 0$ is a homogeneous differential equation. Choose the correct answer from the options given below:

The differential equation of the family of curves $y = Ae^{3x} + Be^{-3 x}$, where $a$ and $\beta$ are arbitrary constants, is

Match List-I with List-II | List-I | List-II | |---|---| | (A) Degree of the differential equation $\frac{d^2y}{dx^2} = e^{dy/dx}$ is | (I) 2 | | (B) Order of the differential equation $(\frac{dy}{dx})^2 + \frac{d^3y}{dx^3} = 0$ is | (II) not defined | | (C) Degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 5x^2 = 0$ | (III) 3 | | (D) If p is the order and q is the degree of the differential equation $\frac{dy}{dx} + 3y = e^x$, then p + q is | (IV) 1 | Choose the correct answer from the options given below:

The solution of the differential equation $(x + 1)\frac{dy}{dx} + 1 - 2e^{-y} = 0$, $y(0) = 0$ is

Match List-I with List-II | List-I | List-II | |---|---| | (A) Integrating factor of $xdy - (y + x^2)dx = 0$ | (I) $x^2$ | | (B) Integrating factor of $xdy + (2y + x^2)dx = 0$ | (II) $x^3$ | | (C) Integrating factor of $(3y - x^2)dx + xdy = 0$ | (III) $x$ | | (D) Integrating factor of $(y + 3x^2)dx + xdy = 0$ | (IV) $\frac{1}{x}$ | Choose the correct answer from the options given below:

The solution of the differential equation $\frac{dy}{dx} = \frac{ax + c}{by + d}$ represents a circle when

Match List-I with List-II | List-I | List-II | | --- | --- | | Differential Equation | General solution | | --- | --- | | (A) $\dfrac{dy}{dx} = \dfrac{y}{x}; x \neq 0$ | (I) $y = cx; c \text{ is an arbitrary constant}$ | | (B) $x dx - y dy = 0; y \neq 0, x \neq 0$ | (II) $x^2 - y^2 = c; c \text{ is an arbitrary constant}$ | | (C) $\dfrac{(x^2 - 1)}{y^2 + 1}\dfrac{dx}{dy} = 1$ | (III) $2x + 3y = c; c \text{ is an arbitrary constant}$ | | (D) $2 dx + 3 dy = 0$ | (IV) $(x^3-y^3) = c + 3(x+y); c \text{ is an arbitrary constant}$ | Choose the correct answer from the options given below:

The number of arbitrary constants in the general solution of a differential equation with degree 1 and order 3, is

A integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2}$, $(x > 0)$ is equal to

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

The differential equation representing the curve $y = e^{2x}(a + bx)$, where a, b are arbitrary constants is

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equation | Order and degree of differential equation | | (A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$ | (I) Order = 1, Degree = 2 | | (B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$ | (II) Order = 2, Degree = 1 | | (C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$ | (III) Order = 2, Degree = 2 | | (D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$ | (IV) Order = 2, Degree = Not defined | Choose the correct answer from the options given below:

The general solution of the differential equation $\frac{dy}{dx} = e^{ax+by}$ is: (Here C is an arbitrary constant)

Solution of the differential equation $\frac{dy}{dx} = \sqrt{1 + x^2 + y^2 + x^2y^2}$ is : (Here $C$ is an arbitrary constant)

Match List-I with List-II | List-I | List-II | |---|---| | Differential Equation | Integrating Factor | | (A) $y dx + (x - y^3)dy = 0$ | (I) $e^{-x}$ | | (B) $x\frac{dy}{dx} + y = x^2$ | (II) $\frac{1}{x}$ | | (C) $\frac{dy}{dx} - y = e^x$ | (III) $y$ | | (D) $x dy - y dx = x^3 dx$ | (IV) $x$ | Choose the correct answer from the options given below:

If the slope of the tangent to the curve $y = y(x)$ at any point $(x, y)$ is $\frac{2x}{y^2}$ and the curve passes through the point $\left(\frac{1}{\sqrt3}, 1\right)$, then equation of curve is

The degree of the differential equation $\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$ is:

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)= 3x + 4y$ satisfying $y = 0$ when $x = 0$ is:

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true? (A) Integrating factor = tan⁻¹x (B) Integrating factor = tan⁻¹y (C) Integrating factor = $e^{tan⁻¹y}$ (D) Degree = 1 Choose the correct answer from the options given below:

The general solution of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

Match List-I with List-II | List-I | List-II | | :--- | :--- | | **Differential equation** | **Order and degree** | | (A) $(y'')^3 + (y')^4 - 6 = (y''')^2$ | (I) Order = 1, Degree = 2 | | (B) $\sqrt{(y')^2 + 5} = y''$ | (II) Order = 2, Degree = 3 | | (C) $(y')^2 = (2 + y'')^{3/2}$ | (III) Order = 2, Degree = 2 | | (D) $y = xy' + \sqrt{a^2(y')^2 + b^2}$ | (IV) Order = 3, Degree = 2 | Choose the correct answer from the options given below:

Consider the differential equation $xdy = (x + y) dx$. Which of the following are true? (A) It is a homogenous differential equation (B) It is a differential equation of order 2 (C) The general solution of the differential equation contains 2 arbitrary constants (D) Integrating factor of differential equation is $\frac{1}{x}$ (E) Degree of the differential equation is not defined Choose the correct answer from the options given below:

The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$. If it passes through the point $(0,0)$, then which of the following is/are true? (A) it is a straight line. (B) it is a parabola. (C) it also passes through the point $(-1, \frac{1}{\sqrt{e}} - 1)$ (D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$ Choose the **correct** answer from the options given below:

The general solution of the differential equation $(x^2 - yx^2)dy + (y^2 + x^2y^2)dx = 0$ is:

The general solution of the differential equation $(1 + e^x)dy + ye^x dx = 0$, where $y > 0$, is

Match List-I with List-II | List-I | List-II | |---|---| | (Differential equation) | (Order and Degree) | | (A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$ | (I) order = 3, degree = 1 | | (B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$ | (II) order = 3, degree not defined | | (C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$ | (III) order = 2, degree = 3 | | (D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$ | (IV) order = 2, degree = 1 | Choose the correct answer from the options given below:

For $|x| < 1$, if $x = \cos\left(\frac{1}{a}\log y\right)$, then

Consider the differential equation $\frac{dy}{dx} + y \tan x = \sec x$, then which of the following statements are correct? (A) It is homogeneous (B) It has $\sec x$ as its integrating factor (C) It's general solution is $y \sec x = \tan x + c$, where c is arbitary constant. (D) It's degree is not defined Choose the correct answer from the options given below:

For $y \neq 0$, the particular solution of the differential equation $2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0$ at the point (1, 1) is

Let $y(x) = a(x + 1) \log(x + 1) + bx + 5$ be the solution of the differential equation e$^\frac{dy}{dx} = x + 1_{;}y(0) = 5$, then the value of $(a + b)$ is:

Match **List-I** with **List-II** | List-I | List-II | |---|---| | **Differential Equations** | **Order and degree** | | (A) $\frac{dy}{dx} + e^y = 0$ | (I) order 2, degree not defined | | (B) $\frac{d^2y}{dx^2} = \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2}$ | (II) order 2, degree 1 | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + e^{(\frac{dy}{dx})} = 0$ | (III) order 1, degree 1 | | (D) $\frac{d^2y}{dx^2} + x\frac{dy}{dx} - 2y = logx; x > 0$ | (IV) order 2, degree 2 | Choose the **correct** answer from the options given below:

Particular solution of the differential equation $x(1 + y^2)dx - y(1 + x^2)dy = 0$, given $y = 0$ when $x = 1$, is

General solution of the differential equation $\frac{dy}{dx} = e^{\frac{x^2}{2}} + xy$ is

For the differential equation $(x + y)dy + (x - y)dx = 0$, which of the following is/are correct? (A) Differential equation is homogeneous (B) Order of differential equation is 1 (C) Integrating factor of differential equation is $e^x$ (D) Degree of the equation is not defined Choose the **correct** answer from the options given below:

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$ is equal to:

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

Which of the following are linear first order differential equations? (A) $\frac{dy}{dx} + P(x)y = Q(x)$ (B) $\frac{dx}{dy} + P(y)x = Q(y)$ (C) $(x - y)\frac{dy}{dx} = x + 2y$ (D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$ Choose the correct answer from the options given below:

Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$, then which of the following are true? (A) It is a linear differential equation (B) It is a homogenous differential equation (C) Its general solution is $\log_e\left(\frac{y}{x}\right) = Cx$, where C is constant of integration (D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$, where C is constant of integration (E) If $y(1) = 1$, then its particular solution is $y = x$ Choose the correct answer from the options given below:

The integrating factor of the differential equation $(x log_e x)\frac{dy}{dx} + y = 2log_e x$ is

Let $e^{\alpha y} + e^{\beta y} + \gamma x^2 + \delta \log|x| + C = 0$, where $C \in \mathbb{R}$ be a particular solution of the differential equation $x(e^{2y} - 1)dy + (x^2 - 1)e^ydx = 0$ and passes through the point $(1, 1)$. The value of $(\alpha + \beta + \gamma + \delta - C)$ is

The general solution of the differential equation $\frac{dy}{dx} = xy + x + y + 1$ is

Match List-I with List-II | List-I | List-II | |---|---| | **Differential equation** | **Degree** | | (A) $\frac{d^2y}{dx^2} + \sqrt{\frac{dy}{dx}} - y = 0$ | (I) 6 | | (B) $\sqrt{\frac{d^3y}{dx^3}} - \sqrt[12]{\frac{d^2y}{dx^2}} = 0$ | (II) Not defined | | (C) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} + e^{\frac{dx}{dx}} = x^2$ | (III) 3 | | (D) $\sqrt[3]{\frac{dy}{dx}} - \frac{d^2y}{dx^2} = e^x$ | (IV) 2 | Choose the correct answer from the options given below:

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

The general solution of differential equation $\frac{dy}{dx} = e^{x+y}$ is

The general solution of the differential equation $e^x dy + (y e^x + 2x)dx = 0$ is

Let the degree and order of the differential equation $2x^3\frac{dy}{dx}-5\left(\frac{d^2y}{dx^2}\right)^2=6\left(\frac{dy}{dx}\right)^3$ be $m$ and $n$ respectively. Then (A) m = 2 (B) n = 3 (C) m = 3 (D) mn = 4 Choose the correct answer from the options given below:

The general solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = ax + by$ is

The general solution of the differential equation $ydx - (x + 2y^2)dy = 0$

Match List-I with List-II | List-I | List-II | |---|---| | Differential equation | Integrating factor | | (A) $x\frac{dy}{dx} - y = 2x^2$ | (I) $e^{-y}$ | | (B) $\frac{dy}{dx} + \frac{y}{x} = 2x$ | (II) $\frac{1}{x}$ | | (C) $x\frac{dy}{dx} + 2y = x^2logx$ | (III) $x$ | | (D) $\frac{dx}{dy} - x = y$ | (IV) $x^2$ | Choose the correct answer from the options given below:

The particular solution of the differential equation $e^x\sqrt{1-y^2}dx + \frac{y}{x}dy = 0$, given that $y = 1$, when $x = 0$ is:

The degree of the differential equation $\left(1-\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}=k \frac{d^{2} y}{d x^{2}}$ is :

Choose the correct answer from the options given below: | List-I | List-II | | --- | --- | | (A) Integrating factor of $ x \, dy - (y + 2x^2) \, dx = 0 $ | (I) $ \frac{1}{x} $ | | (B) Integrating factor of $ (2x^2 - 3y) \, dx = x \, dy $ | (II) $ x $ | | (C) Integrating factor of $ (2y + 3x^2) \, dx + x \, dy = 0 $ | (III) $ x^2 $ | | (D) Integrating factor of $ 2x \, dy + (3x^3 + 2y) \, dx = 0 $ | (IV) $ x^3 $ |

If $\sin y=x \sin (a+y)$, then $\frac{d y}{d x}$ is :

For the differential equation $\left(x \log _{e} x\right) d y=\left(\log _{e} x-y\right) d x$ (A) Degree of the given differential equation is $1$. (B) It is a homogeneous differential equation. (C) Solution is $2y \log _{\mathrm{e}} \mathrm{x}+A=\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}$, where $A$ is an arbitrary constant (D) Solution is $2 y \log _{e} x+A=\log _{e}\left(\log _{e} x\right)$, where $A$ is an arbitrary constant Choose the correct answer from the options given below :

In the context of differential equation Match List I with List II | LIST I | LIST II | |---|---| | A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ | I. Not a differential equation | | B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$ | II. Linear first order | | C. $\sin x + y = \cos(x+y)$ | III. Variable separable | | D. $(x+y)\frac{dy}{dx} = 1$ | IV. Homogenous | Choose the correct answer from the options given below:

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

The solution of $y' - y'' = 2x$ is: A. $y = x^2 + 2x + 2$ B. $y = x^2 + 2x + 1$ C. $y = x + 2$ D. $y = x^2 - 2x + 1$ Choose the correct answer from the options given below:

The general solution of $\frac{dy}{dx} = 1 + x^2 + y^2 + x^2y^2$ is: (given that $C$ is the constant of integration)

The differential equation of the family of curves $y = a \sin(bx + c)$, a and c are parameters, is :

The differential equation $y = xp + \sqrt{x^2 p^3 + 4}$ where $p = \frac{dy}{dx}$ is : (A) of order 1 (B) of degree 1 (C) of order 2 (D) of degree 3 Choose the **correct** answer from the options given below :

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

The equation of curve whose slope is given by $\frac{dy}{dx} = x$ and which passes through $\left(1, \frac{5}{2}\right)$ is :

Match List - I with List - II. Match the integrating factors : | List - I (Differential Equation) | List - II (Integrating factor) | |---|---| | (A) $\frac{dy}{dx} + 3y = e^{-2x}$ | (I) $\frac{1}{x}$ | | (B) $x\frac{dy}{dx} + y = 3x^2$ | (II) $e^{-x}$ | | (C) $x\frac{dy}{dx} - y = 3x^2$ | (III) $x$ | | (D) $\frac{dy}{dx} - y = x$ | (IV) $e^{3x}$ | Choose the correct answer from the options given below :

If m and n are respectively the order and degree of the differential equation : $\left(\frac{d^2 y}{dx^2}\right)^5 + 6 \frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}} + \frac{d^3 y}{dx^3} = x^2 + 5$, then :

The degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)\right]^3 = \left(\frac{d^2 y}{dx^2}\right)^2$ is :

Solution of differential equation $x dy - y dx = 0$ respresents :

The sum of order and degree of the differential equation $\frac{\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{5}{2}}}{\frac{d^2y}{dx^2}} = p$ is :

The solution of the differential equation $\frac{dy}{dx} = \frac{6}{x^2}$; $y(1) = 3$ is :

The differential equation whose solution is $Ax^2 + By^2 = 1$ where A and B are arbitrary constant is of : (A) first order and first degree (B) second order and first degree (C) second order and second degree (D) second order Choose the correct answer from the options given below :

Integrating factor of the differential equation $(1 - y^2)\frac{dx}{dy} + xy = ay$ is :

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$, is :

If curve represented by differential equation $x \frac{dy}{dx} + y = e^x$ passes through (1, 1), then $y(-1)$ is :

If $U(x) = x + \sqrt{1 + x^2}$, then solution of the differential equation $\frac{dy}{dx} + \sqrt{\frac{1 + y^2}{1 + x^2}} = 0$, is :

The differential equation representing family of curves $y = e^{-2x}(a \cos x + b \sin x)$, where a and b are arbitrary constant, is :

Consider the differential equation $\frac{dy}{dx} = \frac{y+1}{x+1}$, and $y=0$ when $x=2$. The value of $y$ at $x=3$ is :

If the order and the degree of the differential equation $\left(\frac{dy}{dx}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{5}}$ are O and S respectively, then $S - O$ is equal to

Match List I with List II | List I | List II | |---|---| | A. The number of arbitrary constants in the particular solution of differential equation of order 2 | I. 1 | | B. The number of arbitrary constants in the general solution of differential equation of order 2 | II. 0 | | C. The integrating factor of differential equation $\frac{dy}{dx} + \frac{1}{x}y = 3, x > 0$, is | III. 2 | | D. For differential equation, $x^2 \frac{dy}{dx} + x = xy, x > 0, \lim_{x \to 0^+} y(x)$ is equal to | IV. $x$ | Choose the correct answer from the options given below:

The curve passing through the point $(-1, 1)$, given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$ also passes through the point $\left( k, -\frac{1}{2} \right)$, then

If the solution curve of the differentiable equation $\frac{dy}{dx} + 2y = e^{3x}$, passes through the point $\left(0, \frac{6}{5}\right)$, then the value of $y(\log_e 2)$ is:

The sum of order and degree of differential equation $2x^2 \cdot \left(\frac{d^2y}{dx^2}\right) - 3 \cdot \left(\frac{dy}{dx}\right)^3 + y = 0$ is

Match List I with List II | List I | List II : Order and degree respectively | |---|---| | A. $\frac{dy}{dx} - (x^2+3) = 0$ | I. 2 and 1 | | B. $2x^2 \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0$ | II. 2 and 3 | | C. $y''' + y^2 + e^{y'} = 0$ | III. 1 and 1 | | D. $\left(\frac{ds}{dt}\right)^4 + 3s\left(\frac{d^2s}{dt^2}\right)^3 = 0$ | IV. 3 and not defined | Choose the correct answer from the options given below:

Solution of the differential equation $\frac{dy}{dx} = x + xy - (1 + y)$ is:

The solution of the differential equation $\frac{dy}{dx} = \frac{\lambda^2}{(x+y)^2}$ ($\lambda$ is constant) is:

The solution of differential equation $\sqrt{x+1} - \sqrt{x-1}\frac{dy}{dx} = 0$ is

If the order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \left(1 + \frac{dy}{dx}\right)^{\frac{1}{3}}$ are $a$ and $b$ respectively, then the value of $a^2 + b^2$ is

The integrating factor of differential equation $x\frac{dy}{dx} + 2y = x^2 \log x$ is

General solution of the differential equation $\frac{dy}{dx} + \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} = 0$ is A. $\tan^{-1} x + \tan^{-1} y = C$ B. $\sin^{-1} x - \cos^{-1} y = C$ C. $x\sqrt{1 - y^2} - y\sqrt{1 - x^2} = C$ D. $\sin^{-1} x + \sin^{-1} y = C$ E. $\cos^{-1} x + \cos^{-1} y = C$ (where C is arbitrary constant) Choose the correct answer from the options given below

The differential equation representing family of curves $y = ae^{mx} + be^{nx}$, where $a$ and $b$ are arbitrary constants, is

The solution of the differential equation $(x+1)\frac{dy}{dx} = 1 + y$ is

Order and degree of the differential equation $y\frac{dy}{dx} + \frac{4}{\frac{dy}{dx}} = 5$ are

Solution of the differential equation $(x + xy)dy - y(1 - x^2)dx = 0$ is

The integrating factor of the differential equation $\cos x \frac{dy}{dx} + y\sin x = 1$ is

The order and degree of the differential equation $\left[\left(\frac{d^2y}{dx^2}\right)^2 - 3\right]^{\frac{1}{3}} = 2\left(\frac{dy}{dx}\right)^{\frac{1}{4}}$ are

Integrating factor of the differential equation $\frac{dy}{dx} - \frac{1}{x} y = 1$ is:

The order and the degree of the differential equation $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ are respectively:

Match List I with List II. | List I | List II | |---|---| | A. $\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$ | I. order 3, degree 1 | | B. $\left(\frac{d^2y}{dx^2}\right)^2 = 0$ | II. order 2, degree 2 | | C. $\frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y = 0$ | III. order 2, degree 1 | | D. $\sin\left(\frac{dy}{dx}\right) + 5y = 0$ | IV. order 1, degree is not defined | Choose the correct answer from the options given below:

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

The solution of differential equation $y(1 - x^2)\frac{dy}{dx} = x(1 + y^2)$ is :

The integrating factor of the differential equation $x\frac{dy}{dx} - y = 2x^2$ is :

Match List - I with List - II | List - I (Differential Equation) | List - II (Degree) | |---|---| | A. $\left[1 + (y')^2\right]^2 = y''$ | I. $2$ | | B. $\left[1 + (y'')^3\right]^{\frac{1}{2}} = (y')^3$ | II. $4$ | | C. $(y''')^2 + y'' + 3y' + 5y = e^x$ | III. $1$ | | D. $\left[1 + (y')^3\right]^{\frac{1}{2}} = (y'')^2$ | IV. $3$ | Choose the correct answer from the option given below:

Choose the correct statements: A. The order and degree (if defined) of a differential equation are always positive integrals B. The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation C. If $\frac{dy}{dx} + P(x)y = Q(x)$ then Integrating factor $= e^{\int P(x)dx}$ D. The sum of order and degree of differential equation $1 + (y'')^5 = (y''')^3$ is $8$ E. If the solution of a differential equation of order $n$, contains $n$ arbitrary constant, then it is called a general solution Choose the correct answer from the options given below:

The differential equation $\frac{dy}{dx} + \frac{x}{y} = 0$, represents the family of curves:

The degree of the differential equation $\left(1 + \frac{dy}{dx}\right)^4 = \left(\frac{d^2y}{dx^2}\right)^2$ is:

Solution of $\frac{dy}{dx} = (1+x^2)(1+y^2)$ is:

Particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = x + y$, given that when $x = 0, y = 0$ is:

The solution of the differentiable equation $2x\frac{dy}{dx} + y = 14x^3, x > 0$, is