Differential equations practice · Mixed difficulty

The differential equation of family of the curves $x^2+y^2=2 a x$ is ..... . ( $a$ is an arbitrary constant)

Solution of differential equation $\left(x+3 y^2\right) \frac{d y}{d x}=y,(y>0)$ is

If $\phi \left( x \right) = \phi '\left( x \right)$ and $\phi \left( 1 \right) = 2$, then $\phi \left( 3 \right)$ equals

The equation of curve passing through the point $\left( {1,\frac{\pi }{4}} \right)$ and having slope of tangent at any point $\left( {x,y} \right)$ as $\frac{y}{x} - {\cos ^2}\left( {\frac{y}{x}} \right)$, is

The differential equation of the family $y = a{e^x} + bx{e^x} + c{x^2}{e^x}$ of curves, where $a,b,c$ are arbitrary constants, is

${\text{If}}\,\,y = \sin (m{\operatorname{Sin} ^{ - 1}}x),{\text{then}}\,\,{\text{ < br > }}(1 - {x^2})\frac{{{d^2}y}}{{d{x^2}}} =$

The solution of differential equation $2 x^3 y d y+\left(1-y^2\right)\left(x^2 y^2+y^2-1\right) d x=0$ is:

If $f(x)$ and $g(x)$ are two solutions of the differential equation $a \frac{d^2 y}{d x^2}+x^2 \frac{d y}{d x}+y=e^x$, then $f(x)-g(x)$ is the solution of

A curve that passes through $(2,4)$ and having subnormal of constant length of 8 units can be

The solution of the differential equation $\frac{d y}{d x}=\frac{1}{x y\left(x^2 \sin y^2+1\right)}$ is

If a curve passes through the point $\left(2, \frac{7}{2}\right)$ and has slope $\left(1-\frac{1}{x^2}\right)$ at any point $(x, y)$ on it, then the ordinate of the point on the curve whose abscissa is -2 is :

The solution of the differential equation $\frac{{dy}}{{dx}} = y\tan x - 2\sin x$, is

A curve passes through the point $\left( {0,1} \right)$ and the gradient at $\left( {x,y} \right)$ on it is $y\left( {xy - 1} \right)$. The equation of the curve is

The equation of the curve through the point (3,2) and whose slope $\frac{{{x^2}}}{{y + 1}}$, is

The solution of $\left(1+\sin ^2 x\right) d y-\cos x\left(1+y^2\right) d x=0$ given that $y=2$ when, $x=\frac{\pi}{2}$

$\int \frac{\left(x^4-x\right)^{1 / 4}}{x^5} d x$ is equal to : (where $\mathrm{c}$ is constant of integration)

The solution of the differential equation $\cos y d x+\left(1+2 e^{-x}\right) \sin y d y=0$, where $y(0)=\frac{\pi}{4}$, is

The solution of the differential equation $\left(1+y^2\right)+\left(x-e^{\operatorname{tan} x^{-1} y}\right) \frac{d y}{d x}=0$ is ...... .

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

Which of the following is true for y(x) that satisfies the differential equation $\frac{{dy}}{{dx}} = \mathrm{ }xy\mathrm{ }-\mathrm{ }1\mathrm{ } + \mathrm{ }x\mathrm{ }-\mathrm{ }y;\mathrm{ }y\mathrm{ }\left( 0 \right)\mathrm{ } = 0$

The rate of decay of a radioactive body at any time is proportional to its mass present at that time. If the mass of body is $100 \mathrm{gms}$ after one hour and $80 \mathrm{gms}$ after two hours, then the initial mass of radioactive body is ........ .

The order of the differential equation of the family of rectangular hyperbolas having fixed centre, is

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

Let $y=y(x)$ be the solution of the differential equation $\cos x \frac{d y}{d x}+y \sin x=1, x \in\left(0, \frac{\pi}{2}\right)$. If ${y}\left(\frac{\pi}{4}\right)=0$, then ${y}\left(\frac{\pi}{6}\right)$ is

The solution of $\frac{{dy}}{{dx}} + y = {e^{ - x}},y\left( 0 \right) = 0$ is

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

If $y:[1, \infty) \rightarrow[1, \infty)$ is a function of $\mathrm{x}$ such that $2 x=y^{1 / 5}+y^{-1 / 5}$ and $\left(x^2-1\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}=k y(k \in R)$ Then $\mathrm{k}=$

Solution of the differential equation $\frac{d y}{d x}=\frac{y^3+y^2 \sqrt{y^2-x^2}}{x^3}$ is

Suppose $y$ is a function of $x$ that satisfies $\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{x^2}$ and $y=0$ at $x=\frac{2}{\pi}$, then $y\left(\frac{3}{\pi}\right)$ is equal to:

The solution of the differential equation $\frac{d y}{d x}+y \cot x=2 \cos x$ is