Variable Separable Form

The value of  ${\int }_{-2}^2|1-x^2|dx$  is:

Solve the differential equation $\frac{dy}{dx}+2xy=y; y\left(0\right)=1$ and express the result in the form of $y=f\left(x\right)$, hence find $f\left(2\right)$.

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

At any point on the curve $y=f\left(x\right)$ the length of the sub-normal is constant, then the curve is

Consider a polynomial function $y=P\left(x\right)$ of least degree, passing through $A\left(-1,1\right)$ and whose graph has two points of inflections $B\left(1,1\right)$ & $C\left(0,a\right)$. At point $\mathrm{C}$, the slope of the tangent drawn to the curve $y=P\left(x\right)$ is $1$. Then which of the following is/are true ?

A river of width $a$ is flowing towards positive $x$ axis. The velocity of current is directly proportional to product of distances from the two banks, constant of proportionality being $k$. A boat is rowed with a velocity $u$ directly across the stream in positive $y$ direction. (Assume starting point as origin ) which of the following is true for the trajectory of the boat.

The general solution of the differential equation $ydy+\sqrt{1+y^2}dx=0$ represents a family of

Solve:

$\frac{dy}{dx}=\frac{4x+6y+5}{3y+2x+4}$

Solution of the following differential equation

$\frac{dy}{dx}=x\sqrt{25-x^2}$ is

The solution of the following differential equation $y-x\frac{dy}{dx}=0$ is

The particular solution of the differential equation: 

$y(1+\log x)\frac{dx}{dy}-x\log x=0$ when $y=e^2$ and $x=e$ is 

Find the equation of a curve passing through the point $(2,3)$, given that the slope of the tangent to the curve at any point $(x,y)$ is $\frac{2x}{y^2}$.

Find the equation of the curve that passes through the point $(1,2)$ and has at every point $\frac{dy}{dx}=\frac{-2xy}{x^2+1}$

Solve the differential equation.

$y-x\frac{dy}{dx}=a\left(y^2+\frac{dy}{dx}\right)$

Solve the differential equation $e^{\frac{dy}{dx}}=x$.

Solve the differential equation.

$\frac{dy}{dx}=1+x+y+xy$

Reduce of the following differential equation to the variables separable form and hence solve.

$\left(x\frac{dy}{dx}-y\right)e^{\frac{y}{x}}=x^2\cos x$

Reduce the following differential equation to the variables separable form and hence solve.

$\frac{dy}{dx}={\left(4x+y+1\right)}^2$

Reduce of the following differential equation to the variables separable form and hence solve.

$x+y\frac{dy}{dx}=\sec \left(x^2+y^2\right)$

Reduce the following differential equation to the variables separable form and hence solve.

$1+\frac{dy}{dx}=\mathrm{cosec}\left(x+y\right)$