Answer to Question #275180 in Differential Equations for Adnan

Question

Find two members of the family of solution in problem 5 that satisfy the initial condition y(0)=0, y'(0)=0.

Answer 1

Given that $y = c_1 + c_2x^2$  is a two-parameter family of solutions of $xy''-y'=0$ on the interval $(-∞,∞).$

$y'=2c_2x$

$y(0)=0:0=c_1 + c_2(0)^2=>c_1=0$

$y'(0): 0=2c_2(0)=>c_2\in \R$

$y=c_2x^2$

Let $c_2=1,$ then $y=x^2.$

Let $c_2=-2,$ then $y=-2x^2.$