# Solutions of the differential equation $x^2y’’-4xy’+6y=0$.

ordinary differential equations

In one of my test it given to prove that $$x^3$$ and $$x^2|x|$$ are linear independent solutions of the differential equation $$x^2y’’-4xy’+6y=0$$ on $$\mathbb R$$( here $$x$$ is independent variable).

But according to me it’s Cauchy Euler equation having general solution as $$y=c_1x^3+c_2x^2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants. How can be $$x^2|x|$$ a solution of given ODE as I am unable to find its by giving particular values of constants $$c_1$$ and $$c_2$$? Please help me to solve it . Thank you.

The differential equation has a singularity at $$x=0$$, so the Existence and Uniqueness Theorem doesn't apply there. On each of the intervals $$(-\infty, 0)$$ and $$(0,\infty)$$ where the theorem does apply, you have two-parameter families of solutions. But it turns out any solution on $$(-\infty, 0)$$ and any solution on $$(0,\infty)$$ with the same $$c_2$$ can be put together to make a solution on $$\mathbb R$$.