Differential equations practice · Mixed difficulty

Let $u(x)$ and $v(x)$ are differentiable functions such that $\frac{u(x)}{v(x)}=7$. If $\frac{u^{\prime}(x)}{v^{\prime}(x)}=p$ and $\left(\frac{u(x)}{v(x)}\right)=\mathrm{q}$, then $\frac{p+q}{p-q}$ has the value equal to :

Solution of the equation ${x^2}y - {x^3}\frac{{dy}}{{dx}} = {y^4}\cos x$, when $y\left( 0 \right) = 1$ is

The solution of the differential equation $\frac{{dy}}{{dx}} = \frac{{x - y + 3}}{{2\left( {x - y} \right) + 5}}$ is

The differential equation $\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}$ determines a family of circles with

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

The order and the degree of $2 x=e^{\frac{d y}{d x}}$ are ..... and ......

The solution of initial value problem $x y^{\prime}+y$ $=0, y(2)=-2$ is given by

The solution of differential equation $2 x y \sec ^2\left(y^2\right) \frac{d y}{d x}+y^4=\ln \left(x^2 e^{y^4}\right)$ is

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

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

The differential equation $\frac{{dy}}{{dx}} = \frac{{x\left( {1 + {y^2}} \right)}}{{y\left( {1 + {x^2}} \right)}}$ represents a family of

If the solution of differential equation $\frac{d y}{d x}=\frac{m x+5}{4 y+f}$ represents a circle, then the values of $m\, \&$ radius $r$ of circle respectively satisfies

The general solution of the differential equation $y d x-x d y-3 x^2 y^2 e^{x^3} d x=0$ is ......

Solution of the differential equation, $\sin y \frac{d y}{d x}=\cos y(1-x \cos y)$ is

The solution of the equation $y^{\prime}=\frac{y+1}{x}$ is given by

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

The solution of the differential equation $x \frac{d^2 y}{d x^2}=1$, given that $y=1, \frac{d y}{d x}=0$ when $x=1$, is

Solution of differential equations $(x \cos x-\sin x) d x=\frac{x}{y} \sin x d y$ is

Let $y = y(x)$ be the solution of the differential equation $\frac{{dy}}{{dx}} = 2(y + 2\sin x - 5)x - 2\cos x$ such that, $y(0) = 7.$ Then $y(\pi )\,$ is equal to

The differential equation of the family of ellipse having major and minor axes respectively along the $x$ and $y$-axes and the minor axis is equal to half of the major axis, is

For solving $\frac{{dy}}{{dx}} = 4x + y + 1$, suitable substitution is

Solution of the differential equation $x\mathrm{ }dy - y\mathrm{ }dx = 0$ represents

If $\frac{{dy}}{{dx}}\, = \,\frac{{{2^{x + y}} - {2^x}}}{{{2^y}}},\,y(0)\, = \,1$ then $y(1)$ is equal to

The particular solution of the differential equation $\frac{d y}{d x}+\frac{2 y}{x}=0$ with initial condition $y(1)=1$ is

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

Which of the following equation is linear

The differential equation of all circles of radius $a$ is of order

A curve $y = f\left( x \right)$ passes through the point $P\left( {1,1} \right)$. The normal to the curve at point $P$ is $a\left( {y - 1} \right) + \left( {x - 1} \right) = 0$. If the slope of the tangent at any point on the curve is proportional to the ordinate at that point, then the equation of the curve is

If $y d x+\left(x+x^2 y\right) d y=0$, then $x=$_____

The differential equation $y\frac{{dy}}{{dx}} + x = a$ ($a$ is any constant) represents