Differential equations practice · Mixed difficulty

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

Let $y=y(x)$ be the solution curve of the differential equation $\sec y \frac{d y}{d x}+2 x \sin y=x^3 \cos y,$ $y(1)=0.$ Then $y(\sqrt{3})$ is equal to:

If $\sin x$ is an intergrating factor of the differential equation $\frac{d y}{d x}+P y=Q,$ then $P$ can be

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

If $y = y\left( x \right)$ is the solution of the differential equation, $x\frac{{dy}}{{dx}} + 2y = {x^2}$ satisfying $y(1) = 1,$ then $y\left( {\frac{1}{2}} \right)$ is equal to

If $y=y(x), x \in\left(0, \frac{\pi}{2}\right)$ be the solution curve of the differential equation $\left(\sin ^2 2 x\right) \frac{d y}{d x}+\left(8 \sin ^2 2 x+\right.$ $2 \sin 4 x) y=2 e^{-4 x}(2 \sin 2 x+\cos 2 x)$, with $y\left(\frac{\pi}{4}\right)=e^{-\pi}$ then $y\left(\frac{\pi}{6}\right)$ is equal to

An equation of the curve for which the portion of $y$-axis cut off between the origin and the tangent varies as the cube of the abscissa of the point of contact is ( $K$ is constant of proportionality)

The order and degree of the differential equation $\sqrt {y + \frac{{{d^2}y}}{{d{x^2}}}} = x + {\left( {\frac{{dy}}{{dx}}} \right)^{\frac{3}{2}}}$ are

General solution of the differential equation $\frac{{dy}}{{dx}} = \frac{{x + y + 1}}{{x + y - 1}}$ is given by

The solution of the equation $x \frac{d y}{d x}+3 y=x$ is

The solution of the differential equation $x{y^2}dy - \left( {{x^3} + {y^3}} \right)dx = 0$ is

Let $f(x), g(x)$ be twice differentiable function on $[0,2]$ satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x), f^{\prime}(1)=2 g^{\prime}(1)=4$ and $f(2)=3, g(2)=9$, then the value of $f(4)-g(4)$ is

If $y = y(x)$ is the solution curve of the differential equation ${x^2}dy + \left( {y - \frac{1}{x}} \right)dx = 0;$ $x > 0\mathrm{ }\,\,\mathrm{and }\,\,y(1) = 1,\,\,\,\mathrm{then}\,\,\mathrm{ }y\left( {\frac{1}{2}} \right)$ is equal to

Let $y=y_1(x)$ and $y=y_2(x)$ be two distinct solution of the differential equation $\frac{d y}{d x}=x+y$, with $y_1(0)=0$ and $y_2(0)=1$ respectively. Then, the number of points of intersection of $y=y_1(x)$ and $y=y_2(x)$ is

Solution of $(x y \cos x y+\sin x y) d x+x^2 \cos x y d y=0$ is

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^2\right) e^{\tan x} d x+\cos ^2 x\left(1+e^{2 \tan x}\right) d y=0$, $y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to :

The degree and order respectively of the differential equation of all parabolas whose axis is $x$-axis, are:

The solution of $d y=\cos x(2-y \operatorname{cosec} x) d x$ where $y=2$ when $x=\frac{\pi}{2}$ is

The numbers P,Q and R for which the function $f\left( x \right) = P{e^{2x}} + Q{e^x} + Rx$ satisfies the conditions $f\left( 0 \right) = - 1,\,{f^1}\left( {\log 2} \right) = 31$ and $\int_0^{\log 4} {\left[ {f\left( x \right) - Rx} \right]dx=\frac{{39}}{2}}$ are given by

If the area of the figure bounded by a curve, the $x$-axis, and two ordinates, one of which is constant, the other variable is equal to the ratio of the cube of the variable ordinate to the variable abscissa, then the curve is

Let $f$ be a differentiable function such that $f'\left( x \right) = 7 - \frac{3}{4}\frac{{f\left( x \right)}}{x},\,\,\left( {x > 0} \right)$ and $f\left( 1 \right) \ne 4$. Then, $\lim\limits_{x \to {0^ + }} x\,f\left( {\frac{1}{x}} \right)$

A curve passes through the point $(1, \pi / 6)$. Let the slope of the curve at each point $(x, y)$ be $\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0$. Then, the equation of the curve is

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

Solution of the differential equation $\frac{d x}{x}+\frac{d y}{y}=0$, is

The degree of the differential equation corresponding to the family of curves $y = a{\left( {x + a} \right)^2}$, where a is an arbitrary constant is

A curve such that the area of the trapezoid formed by coordinate axes, ordinate at an arbitrary point and the tangent at this point equals half the square of its abscissa is

The solution of the equation $x \frac{d y}{d x}=y-x \tan \left(\frac{y}{x}\right)$ is

Which one of the following is a differential equation of the family of curves $y=A e^{2 x}+B e^{-2 x}$

Let $y=y(x)$ be the solution of the differential equation $x \log _e x \frac{d y}{d x}+y=x^2 \log _e x,(x>1)$. If $y(2)=2$, then $y(e)$ is equal to

Solution of differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\frac{1}{(1+\ell \mathrm{nx}+\ell \mathrm{ny})}$ is (where $\mathrm{C}$ is an integration constant)