Prove that if y= sinx/x, show that (d^2y)/(dx^2) + 2/x dy/dx + y = 0 ?

May 13, 2018

See below

Explanation:

If $x y = \sin x$, using the product rule:

• $y + x y ' = \cos x q \quad q \quad q \quad \square$

• $y ' + y ' + x y ' ' = - \sin x q \quad \triangle$

$\frac{\triangle}{x} + \square \implies$

$\frac{2 y '}{x} + y ' ' + \textcolor{red}{y + x y '} = - \textcolor{b l u e}{\frac{\sin x}{x}} + \cos x$

As $\textcolor{b l u e}{y = \frac{\sin x}{x}}$ and $\textcolor{red}{y + x y ' = \cos x}$:

• $\frac{2 y '}{x} + y ' ' + \cos x = - y + \cos x$

• $\implies y ' ' + \frac{2 y '}{x} + y = 0$