CUET Mathematics: CalculusDifferential Equations. Free, no login required.

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) $(1 + x^2)\frac{dy}{dx} - 2xy = x^2 + 3$

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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 $x = 3y^2 - 1$

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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

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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.

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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

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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

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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 :

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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

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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.

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