Differential equations practice · Mixed difficulty

The solution of the differential equation $9y\frac{{dy}}{{dx}} + 4x = 0$ is

The solution of the differential equation $\frac{d y}{d x}=\frac{x^2+y^2+1}{2 x y}$ satisfying $y(1)=1$, is

Differential equation whose solution is $y=c x+c-c^3$, is

The solution of $\frac{{dy}}{{dx}} = 1 + y + {y^2} + x + xy + x{y^2}$ is

If $\phi(x)$ is a differentiable function then the solution of $\mathrm{d} y+\left(y \phi^{\prime}(x)-\phi(x) \phi^{\prime}(x)\right) \mathrm{d} x=0$ is

The differential equation obtained by eliminating arbitrary constants from $y = a{e^{bx}}$ is

Differential equation of all circles with centres on y-axis is

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

Let the solution curve of the differential equation $x \frac{d y}{d x}-y=\sqrt{y^2+16 x^2}, y(1)=3$ be $y=y(x)$. Then $y(2)$ is equal to :

The integrating factor of the differential equation $\left( {y\log y} \right)dx = \left( {\log y - x} \right)dy$ is

The solution of differential equation $y\left(2 x^4+y\right) d y+\left(4 x y^2-1\right) x^2 d x=0$ (where ${C}$ is an arbitrary constant)

Let $y = y\left( x \right)$ be the solution of the differential equation, $\frac{{2 + \sin x}}{{y + 1}}.\frac{{dy}}{{dx}} = - \cos x,y > 0,\,y\left( 0 \right) = 1$. If $y\left( \pi \right) = a$ and $\frac{{dy}}{{dx}}$ at $x = \pi$ is $b$, then the ordered pair $\left( {a,b} \right)$ is equal to

The solution of the differential equation

$x^2 \frac{d y}{d x} \cdot \cos \frac{1}{x}-y \sin \frac{1}{x}=-1,$

where $y \rightarrow-1$ as $x \rightarrow \infty$ is

${y}={f}({x})$ passes through $(1,2)$ and satisfies the relation ${y}(1+x y) d x-x d y=0$. Find value of $7 f\left(\frac{1}{2}\right)$.

Solution of the differential equation $(x-y)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}=$ $a^2$ is

The solution of differential equation $\frac{d y}{d x}=\frac{y}{2 y\left(\sin y^2+\cos y^2\right)-x}$ is

If $y(x)$ is the solution of the differential equation $(x \log x) \frac{d y}{d x}+y=2 x \log x(x \geq 1)$, then $y(e)=\ldots \ldots$

Differential equation whose sol. is $y=c x+c-c^3$, is

If $y = f\left( x \right)$ is the equation of the curve an its differential equation is given by $\frac{{dy}}{{dx}} = \frac{{x + 2}}{{y + 3}},$ then the equation of the curve, if it passes through (2,2), is

General solution of the equation $\left(1+x^2\right) \frac{d y}{d x}+y=e^{\operatorname{Tan}^{-1} x}$ is

The equation to the orthogonal trajectories of the system of parabolas $y=a x^2$ is

The solution of the differential equation $x^3 \frac{d y}{d x}+4 x^2 \tan y=e^x \sec y$ satisfying $y(1)=0$, is

The differential equation obtained on eliminating $A$ and $B$ from the equation $y = A\cos \omega t + B\sin \omega t$ is

Solution of the differential equation, $y d x-x d y+x y^2 d x=0$ can be

For the differential equation $\frac{d^2 y}{d x^2}+y=0$, let its solution be
$y=\phi_1(x)=2 \sin \left(x+\frac{\pi}{4}\right)$.

Assertion: The function $\mathrm{y}=\phi_1(\mathrm{x})$ is called the particular solution.

Reason: The solution which is free from arbitrary constant, is called a particular solution.

If $y=f(x)$ satisfies the differential equation $\sin x \frac{d y}{d x}+2 y \cos x=8$ with $f(\pi / 2)=8$, then the minimum value of $f(x)$ is:

The integrating factor of the differential equation $\frac{{dy}}{{dx}} + \frac{1}{x}.y = 3x$ is

A particular solution of the initial value differential equation $\log \frac{d y}{d x}=7 x+8 y, y(0)=0$ is

Let $y=y(x)$ be the solution of the differential equation $\sec x d y+\{2(1-x) \tan x+x(2-x)\} d x=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

Consider the differential equation, ${y^2}dx + \left( {x - \frac{1}{y}} \right)dy = 0.$ if value of $y$ is 1 when x=1, then the value of $x$ for which $y = 2$ is