Differential equations practice · Easy

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

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

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

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

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

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.

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

The solution of the differential equation $y^3 x^2 \cos \left(x^3\right) d x+\sin \left(x^3\right) y^2 d y=\frac{x}{3} d x$ is $2 \sin \left(x^3\right) y^k=x^2+C$ (where ${C}$ is an arbitrary constant) then the value of ${k}$ is equal to

The order and degree of the differential equation $\sqrt {\sin x} \left( {dx + dy} \right) = \sqrt {\cos x} \left( {dx - dy} \right)$ are

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

The solution of the differential equation $\frac{d y}{d x}+\frac{y}{x}=x^2$ is

The general solution of the differential equation $100\frac{{{d^2}y}}{{d{x^2}}} - 20\frac{{dy}}{{dx}} + y = 0$ is

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

Let $f$ be a twice differentiable function defined on $R$, such that $f(0) = 1,\,\,\,\,f'(0) = 2$ and $f''(x) \ne 0$ for all $x \in R.$ If $\left| {\begin{array}{cccccccccccccccccccc} {f(x)}&{f'(x)}\\ {f'(x)}&{f''(x)} \end{array}} \right| = 0$ for all $x \in R$, then the value of $f(1)$ lies in the interval

If $y=e^x(\sin x+\cos x)$, then the value of $\frac{d^2 y}{d x^2}-2 \frac{d y}{d x}+2 y$, is

The family of curves $y = {e^{a\sin x}}$, where $a$ is an arbitrary constant, is represented by the differential equation

If $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{3}{\cos ^2 x} y=\frac{1}{\cos ^2 x}, x \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$, and $y\left(\frac{\pi}{4}\right)=\frac{4}{3}$, then $y\left(-\frac{\pi}{4}\right)$ equals:

If $p$ and $q$ are the order and degree of the differential equation $y \frac{d y}{d x}+x^3 \frac{d^2 y}{d x^2}+x y=\cos x$, then

The solution of the differential equation $\frac{d y}{d x}=1+x+y+x y$ is:

Equation of the curve through the origin satisfying $\mathrm{d} y=(\sec x+y \tan x) \mathrm{d} x$ is

The solution of the differential equation $\frac{d y}{d x}+\frac{3 x^2}{1+x^3} y=\frac{\sin ^2 x}{1+x^3}$ is

The solution of $y^{\prime}+\cos (x+y)=\cos (x-y)$ and $y\left(\frac{\pi}{4}\right)=\frac{\pi}{4}$ is given by

Let $y(x)$ be a solution of $\frac{(2+\sin x)}{(1+y)} \frac{d y}{d x}=\cos x$. If $y(0)=2$, then $y\left(\frac{\pi}{2}\right)$ equals

The order and degree of the differential equation ${\left( {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right)^{\frac{3}{4}}} = {\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^{\frac{1}{3}}}$ is

The solution of the differential equation $2x\frac{{dy}}{{dx}} - y = 3$ represents

The degree and order of differential equation of the family of all parabolas whose axis is $x$-axis are respectively