Differential equations practice · Mixed difficulty

A solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{x y\left[x^2 \sin y^2+1\right]}$ is
( $C$ is an arbitrary constant)

The elimination of the arbitrary constant $m$ from the equation $y=e^{m x}$ gives the differential equation

If, $x = A\cos 4t + B\sin 4t$ then $\frac{{{d^2}x}}{{d{t^2}}}$ is equal to

The degree of the differential equation $y_3^{2/3} + 2 + 3{y_2} + {y_1} = 0$, is

General solutions of $y^{\prime}=e^{y / x}+y / x$ is

Which of the following statements on ordinary differential equations is/are true?
(i) The number of arbitrary constants is same as the degree of the differential equation.
(ii) A linear differential equation can contain products of the dependent variable and its derivatives.
(iii) A particular integral cannot contain arbitrary constants.
(iv) By putting $v=\frac{y}{x}$ any homogeneous first order differential equation transforms to variable separable form.

The solution of the differential equation $\frac{d y}{d x}=\frac{y \cos x-y^2}{\sin x}$ is equal to (where ${c}$ is an arbitrary constant)

The differential equation of the family of lines through the origin, is

If $\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}$ is the solution of the differential equation $x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x$ and $y(1)=\frac{\pi}{3}$, then $\alpha^2$ is equal to

$\left(x^2+y^2\right) d y=x y d x$. If $y\left(x_0\right)=e, y(1)=1$, then value of $x_0=$

The number of solutions of $y' = \frac{{y + 1}}{{x - 1}},y\left( 1 \right) = 2$ is

If $y=y(x)$ is a function of $x$ and $y(1)=0$, then the solution of the equation $x \frac{d y}{d x}-\frac{y}{x+1}=x$ is

Solution of the equation $x \sin \left(\frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x$ is

The differential equation of all straight lines touching the circle ${x^2} + {y^2} = {a^2}$ is

Which of the following transformation reduce the differential equation $\frac{\mathrm{d} z}{\mathrm{~d} x}+\frac{z}{x} \log z=\frac{z}{x^2}(\log z)^2$ into the form $\frac{\mathrm{d} u}{\mathrm{~d} x}+u P(x)=Q(x)$

The differential equation of the family of non-vertical lines is .......

If $y = a\sin (5x + c)$, then

If $\frac{d^2 y}{d x^2}=0$, then

Let $y=y(x)$ be the solution of the differential equation $x^{3} d y+(x y-1) d x=0, x>0, y\left(\frac{1}{2}\right)=3-e$. Then $y(1)$ is equal to

A solution of $y=x y^{\prime}-3 y^{\prime 3}$ is

If the curve $y = y(x)$ is the solution of the differential equation
$2({x^2} + {x^{5/4}})dy - y(x + {x^{1/4}})\mathrm{ }dx = 2{x^{9/4}}dx,\,\,x > 0$ which passes through the point $\left( {1,\,\,1 - \frac{4}{3}{{\log }_e}2} \right)$,then the value of $y(16)$ is equal to

The value of the constant ' $m$ ' and ' $c$ ' for which $y=m x+c$ is a solution of the differential equation $D^2 y-3 D y-4 y=-4 x$.

The general solution of differential equation

$\frac{dy}{dx}+\frac{y\ell ny}{x}=\frac{y{{(\ell ny)}^{2}}}{{{x}^{2}}}$

(where $\mathrm{C}$ is an arbitrary constant)

If $y=ce^{\sin ^{-1} x}$, then corresponding to this the differential equation is

If $x\mathrm{ }dy = y\left( {dx + y\mathrm{ }dy} \right),y\left( 1 \right) = 1$ and $y\left( x \right) > 0$, then $y\left( { - 3} \right)$ is equal to

The solution of the differential equation $\left(x^2+y^2\right) d x-5 x y d y=0, y(1)=0$, is:

When $x=0, y=1$ the particular solution of the differential equation $\left(1+y^2\right) d x+\left(1+x^2\right) d y=0$ is

Statement 1: The degrees of the differential equations $\frac{d y}{d x}+y^2=x$ and $\frac{d^2 y}{d x^2}+y=\sin x$ are equal.

Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined.

The integral factor of equation $\left( {{x^2} + 1} \right)\frac{{dy}}{{dx}} + 2xy = {x^2} - 1$ is

Solution of $2 y \sin x \frac{d y}{d x}=2 \sin x \cos x-y^{2} \cos x, x=\frac{\pi}{2}, y=1$ is given by