Question

Using x^2=t,reduce the equation
2xydx+(1-x^2+2y)dy=0
to the linear form and hence solve it

Answer 1

The given differential equation is

$2xy\ d x+(1-x^2+2y)dy=0$ .............(1)

Let $x^2=t$ .

On differentiation, $2x\ dx=dt$

Substituting these in Eq.(1) we get,

$ydt+(1-t+2y)dy=0$ .This is the linear form of Eq.(1).

$ydt+(1+2y)dy-tdy=0\\ ydt-tdy+(1+2y)dy=0$

Dividing by $y^2$ ,

$\frac{ydt-tdy}{y^2}+\frac{1+2y}{y^2}dy=0$

$d(t/y)+\left(\frac{1}{y^2}+\frac{2}{y}\right)dy=0$ since $d(t/y)=\frac{ydt-tdy}{y^2}$

Integrating,

$\int d(t/y)+\int \left(\frac{1}{y^2}+\frac{2}{y}\right)dy=C$ , where $C$ is the constant of integration.

$\frac{t}{y}+\left(-\frac{1}{y} +2\ln y\right)=C$

Substituting $t=x^2,$

$\frac{x^2}{y}-\frac{1}{y}+2\ln y=C$

This is the solution of the differential equation (1).