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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:

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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

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

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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 $y = x$

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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

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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

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

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

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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 :

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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

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In the context of differential equation

Match List I with List II

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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. $y = x^2 - 2x + 1$

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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 :