Differential equations practice · Mixed difficulty

The solution of differential equation $y e^{\frac{x}{y}} d x=\left(x e^{\frac{x}{y}}+y^2\right) d y$ 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 curve satisfying the differential equation $\frac{d y}{d x}=\frac{y\left(x+y^3\right)}{x\left(y^3-x\right)}$ and passing through $(4,-2)$ is

A rumour spreeds through a population of 5000 people at a rate proportional to the product of the number of people who have heard it and the number who have not. Suppose that 100 people initiate the rumour and that a total of 500 people know the rumour after 2 days. How long will it take for half the people to hear the rumour. $(\log 9/\log 49=129 / 229)$.

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

General solution of the differential equation $\left(\cos ^2 x\right) \frac{d y}{d x}+y=\tan x \,\,\,\left(0 \leq x < \frac{\pi}{2}\right)$ is

The solution of $y^5 x+y-x \frac{\mathrm{d} y}{\mathrm{~d} x}=0$ 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

The order and degree of the differential equation ${\left( {1 + 3\frac{{dy}}{{dx}}} \right)^{\frac{2}{3}}} = 4\frac{{{d^3}y}}{{d{x^3}}}$ are

The differential equation representing the family of ellipse having foci either on the $x$-axis or on the $y$-axis centre at the origin and passing through the point $(0,3)$ 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

A curve having the condition that the slope of tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates, is a/an

General solution of the equation $\left(1+x^2\right) \frac{d y}{d x}+x y=x^3 y^3$ is

The solution of $\frac{{d\upsilon }}{{dt}} + \frac{k}{m}\upsilon = - g$ is

If $y = {\left( {x + \sqrt {1 + x} } \right)^n}$, then $\left( {1 + {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} + x\frac{{dy}}{{dx}}$ is

If $x=\sec \theta-\cos \theta$ and $y=\sec ^n \theta-\cos ^n \theta$, then $\left(x^2+4\right)\left(\frac{d y}{d x}\right)^2$ is equal to

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 the differential equation $\frac{{dy}}{{dx}} = \frac{{x - 2y + 1}}{{2x - 4y}}$ is

The solution of differential equation $\frac{{dt}}{{dx}} = \frac{{t\left( {\frac{d}{{dx}}\left( {\mathrm{g}\left( x \right)} \right)} \right) - {t^2}}}{{\mathrm{g}\left( x \right)}}$ 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

The solution of differential equation $\left(x y^2-e^{x^{\frac{1}{3}}}\right) d x-x^2 y d y=0$ is