Answer to Question #129296 in Differential Equations for Hassan Rafique

Question

Y=xdy/dx+1/2(dy/dx)^2

Answer 1

As per the question,

y=x$\frac{dy}{dx}$+$\frac{d^2y}{2dx^2}$

Arranging the above equation as,

$\frac{d^2y}{dx^2}$ +2x$\frac{dy}{dx}$ -2y=0 .....(1)(say)

let y=xt,

we take y=xt since It will fulfill the condition

since we have to reduce Second order differential equation into first order.

differentiate both sides with respect to x,

$\frac{dy}{dx}$ =t

Again differentiating above equation with respect to x,

$\frac{d^2y}{dx^2}$ =$\frac{dt}{dx}$

Putting the above values in equation 1 we get,

$\frac{dt}{dx}$ +2xt-2xt=0

$\frac{dt}{dx}$ =0

so t=constant=k(say)

since t=$\frac{y}{x}$

$so \frac{y}{x}$ =k

y=kx

This is the required solution.