Question

In the following example, verify that the given function is a solution of the corresponding differential equation.

Solution	D.E.
y = xn	${\displaystyle x^2\frac{d^{2}y}{{dx}^2}-n\times \frac{xdy}{dx}+ny=0}$

Answer 1

y = x ⁿ

Differentiating w.r.t. x, we get

${\displaystyle \frac{dy}{dx}=n{x}^{n-1}}$

Again, differentiating w.r.t. x, we get

${\displaystyle \frac{d^{2}y}{{dx}^2}=n(n-1)x^{n-2}}$

∴  ${\displaystyle x^2\frac{d^{2}y}{{dx}^2}-nx\frac{dy}{dx}+ny}$

= n(n-1)x²x^n-2 - nx.nx^n-1+ nxⁿ

= n(n-1)xⁿ - n² xⁿ + nxⁿ

=[n(n-1)-n²+n]xⁿ

= 0

∴ ${\displaystyle x^2\frac{d^{2}y}{{dx}^2}-nx\frac{dy}{dx}+ny=0}$

∴ Given function is a solution of the given differential equation.