Differential equations practice · Easy

The solution of equation $\frac{d y}{d x}=\frac{a x+q}{b y+f}$ represents a circle only if

The differential equation for which ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y = c$ is given by

The differential equation of the family of curves ${y^2} = 4a\left( {x + a} \right)$, is

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

Suppose $y=y(x)$ satisfies the differential equation $y d x+y^2 d y=x d y$. If $y(x) > 0 \forall x \in \mathbf{R}$ and $y(1)=1$ then $y(-3)$ equals

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

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

The 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

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 $y(256)=$

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

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

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

The solution of $\frac{d y}{d x}=\frac{a x+b}{c y+d}$ represents a parabola if

The solution of $\frac{x \mathrm{~d} y}{x^2+y^2}=\left(\frac{y}{x^2+y^2}-1\right)$ $\mathrm{d} x$ is

The solution of the differential equation $\frac{{dy}}{{dx}} = {e^{x + y}}$ is

The value of $\int_0^\pi \sin ^3 x \cos ^2 x d x$ is