Differential equations practice · Easy

Solution of the differential equation $2 y \sin x \frac{\mathrm{d} y}{\mathrm{~d} x}=2 \sin x \cos x-y^2 \cos x$ satisfying $y(\pi / 2)=1$ is given by

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

If $y=(A+B x) e^{m x}+(m-1)^{-2} e^x$, then $\frac{d^2 y}{d x^2}-2 m \frac{d y}{d x}+m^2 y$ is equal to

Let y = y(x) be the solution of the differential equation ${e^x}\sqrt {1 - \mathrm{ }{y^2}} \mathrm{ }dx\mathrm{ } + \left( {\frac{y}{x}} \right)dy\, = \mathrm{ }0,\mathrm{ }y\left( 1 \right)\mathrm{ } = \mathrm{ } - 1.$ Then, the value of ${\left[ {y\left( 3 \right)} \right]^2}$ is equal to

If the solution of the differential equation $\frac{{dy}}{{dx}} + {e^x}\left( {{x^2} - 2} \right)y = \left( {{x^2} - 2x} \right)\left( {{x^2} - 2} \right){e^{2x}}$ satisfies $y\left( 0 \right) = 0,$ then the value of $y\left( 2 \right)$ is_____.

The solution of the differential equation $1-\frac{d y}{d x}=e^{x-y}$ is

Let $y = y(x)$ satisfies the equation $\frac{{dy}}{{dx}} - \left| A \right| = 0$, for all $x > 0$ where $A = \left[ {\begin{array}{cccccccccccccccccccc} y&{\sin x}&1\\ 0&{ - 1}&1\\ 2&0&{\frac{1}{x}} \end{array}} \right]$. If $y(\pi ) = \pi + 2$, then the value of $y\left( {\frac{\pi }{2}} \right)$ is

The solution of $\frac{{dy}}{{dx}} = \frac{{ax + h}}{{by + k}}$ represents a parabola, when

Solution of the equation $x{\left( {\frac{{dy}}{{dx}}} \right)^2} + 2\sqrt {xy} \frac{{dy}}{{dx}} + y = 0$ is

$\frac{d^2 x}{d y^2}=\ldots \ldots$

An integrating factor of the differential equation $\left( {1 + y + {x^2}y} \right)dx + \left( {x + {x^3}} \right)dy = 0$ is

The interval contained in the domain of definition of non-zero solutions of the differential equation $(x-3)^2 y^{\prime}+y=0$, is

The degree and the order of $\left.y=x \frac{d^2 y}{d x^2}+3 \sqrt{1+\left(\frac{d^3 y}{d x^3}\right.}\right)^3$ are and respectively.

A wet porous substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet hung in the wind loses half its moisture during the first hour, then the time when it would have lost $99.9 \%$ of its moisture is (weather conditions remaining same)

The solution of $\frac{d y}{d x}+p(x) y=0$, is

The solution of the equation $\frac{d y}{d x}=\sec (x+y)$ is given by

The degree of differential equation $\left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}=\frac{d^2 y}{d x^2}$ is

Which of the following equation is linear

Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=0$, then $y(2)$ is equal to

The sum of the squares of the perpendicular drawn from the points $(0,1)$ and $(0,-1)$ to any tangent to a curve is 2. The equation of the curve, is

If $y(x)$ satisfies the differential equation $y^{\prime}-y \tan x=2 x \sec x$ and $y(0)=0$, then

Let the solution curve $y=f(x)$ of the differential equation $\frac{d y}{d x}+\frac{x y}{x^2-1}=\frac{x^4+2 x}{\sqrt{1-x^2}}, x \in(-1,1) \quad$ pass through the origin. Then $\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) d x$ is

The solution of $y^{\prime \prime}-4 y^{\prime}+3 y=0, y(0)=6$ and $y^{\prime}(0)=10$ is

The differential equation of all ‘Simple Harmonic Motions’ of given period $\frac{{2\pi }}{n}$, is

The curve passing through $(0,1)$ and satisfying $\sin \left(\frac{d y}{d x}\right)=\frac{1}{2}$ is

Let $\frac{{dy}}{{dx}} = \frac{{ax - by + a}}{{bx + cy + a}},$ where $a,b,c$ are constants, represents a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is

If $\frac{d x}{d y}=\left(e^y-x\right)$, where $y(0)=0$, then $y$ is expressed explicitly as

If $c$ is an arbitrary constant, then the general solution of the differential equation $y\mathrm{ }dx - x\mathrm{ }dy = xy\mathrm{ }dx$ is given by

The solution of $\left(1+y+x^2 y\right) \mathrm{d} x+\left(x+x^3\right) \mathrm{d} y=0$ is

General solution of differential equation $x^2\left(x+y \frac{d y}{d x}\right)+\left(x \frac{d y}{d x}-y\right) \sqrt{x^2+y^2}=0$ is