Differential equations practice · Medium

The differential equation of the family of parabola with focus as the origin and the axis as $x$-axis, is

The differential equation obtained by eliminating the arbitrary constants $a$ and $b$ from $xy = a{e^x} + b{e^{ - x}}$ is

The solution of the differential equation $x\mathrm{ }dy - y\mathrm{ }dx = \sqrt {{x^3} + {y^2}} dx$, is

The differential equation of all non-vertical lines in a plane is

The order of differential equation of all parabola with it’s axis parallel to $y$-axis and touch $x$-axis is

The solution of the equation $\frac{d y}{d x}=\frac{y}{x}\left(\log \frac{y}{x}+1\right)$ is

A solution of $y=2 x\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)+x^2\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^4$ is

Observe the following statements

A:Interating factor of $\frac{{dy}}{{dx}} + y = {x^2}$ is ${e^x}$
R: Integrating factor of $\frac{{dy}}{{dx}} + P\left( x \right)y = Q\left( x \right)$ is ${e^{\int P\left( x \right)dx}}$
Then, the true statement among the following is

The differential equation for all family of lines which are at a unit distance from the origin is:

The solution of the differential equation $\frac{{dy}}{{dx}} = x\log x$ is

The general solution of ${y^2}dx + \left( {{x^2} - xy + {y^2}} \right)dy = 0$ is

The solution of $y_2-2 y_1+y=0$ is

Let $f(x)=\left(\sin \left(\tan ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right)^2-1,|x| > 1$. If $\frac{d y}{d x}=\frac{1}{2} \frac{d}{d x}\left(\sin ^{-1}(f(x))\right)$ and $y(\sqrt{3})=\frac{\pi}{6}$, then $y(-\sqrt{3})$ is equal to:

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

Integrating factor of equation $\left(x^2+1\right) \frac{d y}{d x}+2 x y=x^2-1$ is

The solution of the differential equation $y' = 1 + x + {y^2} + x{y^2},y\left( 0 \right) = 0$ is

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

The solution of the equation $\frac{d y}{d x}+x(x+y)=x^3(x+y)^3-1$ is

Let $y=y(x)$ be the solution of the differential equation $\left(x^{2}-3 y^{2}\right) d x+3 x y \, d y=0, \, y(1)=1$. Then $6 y^{2}(e)$ is equal to

The solution of $y' - y = 1,y\left( 0 \right) = - 1$ is given by $y\left( x \right)$, which is equal to

If $\frac{{dy}}{{dx}} + y = 2{e^{2x}}$, then $y$ is equal to

Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by the differential equation $\frac{d V}{d t}=-k(T-t)$, where $k>0$ is a constant and $T$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is:

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

The length of the subnormal of the curve $x^2=4 y$ at point $(-2,1)$ is

The solution of the differential equation $\left( {1 + {y^2}} \right) + \left( {x - {e^{{\mathop\mathrm{t}\nolimits} {{\tan }^{ - 1}}y}}} \right)\frac{{dy}}{{dx}} = 0$ is

$\frac{d^2 x}{d y^2}$ equals to

The solution of $\frac{{dy}}{{dx}} = {\left( {\frac{y}{x}} \right)^{\frac{1}{3}}}$, is

The solution of the equation $d y / d x=\cos (x-y)$, where $y\left(\frac{3 \pi}{2}\right)=\pi$ is

The equation of the curve whose subnormal is equal to a constant $a$ is

The solution of the D.E $y d x-x d y+3 x^2 y^2 d x=0$ is