Differential equations practice · Mixed difficulty

Let $y=y(x)$ be the solution curve of the differential equation $\frac{d y}{d x}=\frac{y}{x}\left(1+x y^{2}\left(1+\log _{e} x\right)\right), x>0, y(1)=3$. Then $\frac{y^{2}(x)}{9}$ is equal to

The solution of $\frac{d y}{d x}=\frac{a x+b}{c y+d}$ represents a parabola if

Solution of $\frac{d y}{d x}+\sin \left(\frac{x+y}{2}\right)=\sin \left(\frac{x-y}{2}\right)$ is

A curve passing through $(2,3)$ and satisfying the differential equation $\int_0^x t y(t) d t=x^2 y(x),(x>0)$ is

${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = C$ is the general solution of the differential equation

Let $x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}$ and
$y\left( t \right) = 2\sqrt 2 \sin t\sqrt {\sin 2t} ,\,\,\,t \notin \left( {0,\frac{\pi }{2}} \right).$ Then
$\frac{{1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}}}{{\frac{{{d^2}y}}{{d{x^2}}}}}\,at\,\,t = \frac{\pi }{4}$ is eqaul to

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

Solution of the differential equation $y d x+(x+y) d y=0$ is

If $y = y\left( x \right)$ and $\frac{{2 + \sin x}}{{y + 1}}\left( {\frac{{dy}}{{dx}}} \right) = - \cos x,y\left( 0 \right) = 1$ then $y\left( {\frac{\pi }{2}} \right)$ equals

$y+x^2=\frac{d y}{d x}$ has the solution

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

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

Solution of the differential equation $\frac{d y}{d x}=\mathrm{y} \tan x-2 \sin \mathrm{x}$, is

The solution of the equation $\left(x+2 y^3\right) \frac{d y}{d x}-y=0$ is [Where $A$ is any arbitrary constant]

The solution of differential equation $t = 1 + \left( {ty} \right)\frac{{dy}}{{dt}} + \frac{{{{\left( {ty} \right)}^2}}}{{2!}}{\left( {\frac{{dy}}{{dx}}} \right)^2} + ...\infty$ is

The differential equation of the family of curves $y = a\cos \left( {x + b} \right)$ is

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

If $\frac{d y}{d x}+y \cot x=\cos x, y\left(\frac{\pi}{2}\right)=\frac{1}{2}$ then $y\left(\frac{\pi}{6}\right)=$

The differential equation of system of concentric circles with centre (1,2) is

By substituting $y=v x$, the solution of the differential equation $\frac{d y}{d x}=\frac{x^2+y^2}{x y}$, is

The D.E of $x y=a e^x+b e^{-x}+x^2$ is

Let $y=y(x)$ satisfies the equation $\frac{d y}{d x}-|A|=0$ for $x>0$, where $A=\left[\begin{array}{ccc}y & \sin x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{array}\right]$.

If $y(\pi)=\pi+2$ then $y\left(\frac{\pi}{2}\right)=$

Consider the differential equation:
$\frac{d y}{d x}=\frac{y^3}{2\left(x y^2-x^2\right)}$ Statement-1: The substitution $z=y^2$ transforms the above equation into a first order homogenous differential equation.
Statement-2: The solution of this differential equation is $y^2 e^{-y^2} / x=C$

In an experiment of culture of bacteria in a laboratory, the rate of increase of bacteria is proportional to the number of bacteria present at that line. If at the end of one hour the number of bacteria gets doubled, then the number of bacteria at the end of 3 hours is

The solution of $\frac{{dy}}{{dx}} + 2y\tan x = \sin x$, is

The differential equation satisfied by the family of curves $y = ax\cos \left( {\frac{1}{x} + b} \right)$ where $a,b$ are parameters is

For the differential equation $\frac{d y}{d x}+i y=2 \sin (x)$, with initial condition $y(0)=\frac{3}{2}$. If $y(\pi)=\frac{a}{b}+c \pi i$, for co-prime integers $a$ and $b$, find $a b c$. [$i=\sqrt{-1}$ denotes the imaginary unit.]

The equation of the curve passing through $(3,9)$, which satisfies $d y / d x=x+1 / x^2$ is

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

The differential equation for the family of curves $x^2-y^2-2 a y=0$, where $a$ is an arbitrary constant, is