Differential equations practice · Mixed difficulty

If $dx + dy = \left( {x + y} \right)\left( {dx - dy} \right)$, then $\log \left( {x + y} \right)$ is equal to

$y=4 \sin 3 x$ is a solution of the differential equation

I.F. for $y \ln y \frac{d x}{d y}+x-\ln y=0$ is

The equation of the curve satisfying the equation $\left( {xy - {x^2}} \right)\frac{{dy}}{{dx}} = {y^2}$ and passing through the point $\left( { - 1,1} \right)$ is

The order and degree of the differential equation $\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^{\frac{1}{4}}+x^{\frac{1}{5}}=0$, respectively, are

The differential equation of displacement of all "Simple harmonic motions" of given period $2 \pi / n$, is

If $f(x)$ is differentiable function in the interoal $(0, \infty)$ such that $f(1)=1$ and $\lim\limits _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$, then $f\left(\frac{3}{2}\right)$ is equal to

If $y=y(x)$ is the solution of the differential equation, $e^y=e^x$ such that $y(0)=0$, then $y(1)$ is equal to:

The differential equation for which $y=a \cos x+b \sin x$ is a solution, is

Degree of differential equation ${e^{dy/dx}} = x$ is

The differential equation of all circles passing through the origin and having their centres on $x$-axis

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

For solution of $\frac{d y}{d x}+y \tan x=\sec x, y(0)=1$, then $y\left(\frac{\pi}{6}\right)$ is equal to:

Assertion: The differential equation $x^2=y^2+x y \frac{d y}{d x}$ is an ordinary differential equation.

Reason: An ordinary differential equation involves derivatives of the dependent variable with respect to only one dependent variable.

The order of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^3=\left(1+\frac{d y}{d x}\right)^{1 / 2}$

If the solution curve $f(x, y)=0$ of the differential equation $\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$, passes through the points $(1,0)$ and $(\alpha, 2)$, then $a^{a}$ is equal to

An equation of the curve satisfying $x \mathrm{~d} y-y \mathrm{~d} x=\sqrt{x^2-y^2} \mathrm{~d} x$ and $y(1)=0$ is

A function $y=f(x)$ satisfies the differential equation $f(x) \cdot \sin 2 x-\cos x+\left(1+\sin ^2 x\right) f^{\prime}(x)=0$ with initial condition $y(0)=0$. The value of $f(\pi / 6)$ is equal to

If $y=y(x)$ satisfies the differential equation $8 \sqrt{x}(\sqrt{9+\sqrt{x}}) d y=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} d x, x > 0$ and $y(0)=\sqrt{7}$, then $y(256)=$

The solution of the differential equation ${e^{ - x}}\left( {y + 1} \right)dy + \left( {{{\cos }^2}x - \sin 2x} \right)y\mathrm{ }dx = 0$ subjected to the condition that $y = 1$ when $x = 0$ is

Solution of the differential equation $\left\{\frac{1}{x}-\frac{y^2}{(x-y)^2}\right\} d x+\left\{\frac{x^2}{(x-y)^2}-\frac{1}{y}\right\} d y=0$ is

The differential equation of the family of circles with fixed radius 5 unit and centre on the line $y = 2$, is

Solution of differential equation $x d y+y d x=e^{x y-\ell n x^2}(x d y-y d x)$ is given by: (where ${C}$ is arbitrary constant)

A solution of the equation $x \frac{\mathrm{d} y}{\mathrm{~d} x}=y(\log y-\log x+1)$ is

The solution of the differential equation $\left( {1 + {y^2}} \right){\tan ^{ - 1}}xdx + y\left( {1 + {x^2}} \right)dy = 0$ is

Solution of $\frac{d y}{d x}=\frac{x-y+3}{x+y+7}$ is

The differential equation $x \frac{d y}{d x}-y=x^2$, has the general solution

Let $y=y(x), y>0$, be a solution curve of the differential equation $\left(1+x^{2}\right) d y=y(x-y) d x$. If $y(0)=1$ and $y(2 \sqrt{2})=\beta$, then

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

Solution of the differential equation $\ln \left(\frac{d y}{d x}\right)=a x+$ by is