Using substitution to reduce a differential equation to a known form

A substitution can often change a second order differential equation into a linear second order differential equation that can be easily solved. In this video I illustrate this with the following example.

By using the substitution $x = \sqrt t$ .Show that the equation $\dfrac{{{d^2}y}}{{d{x^2}}} - \dfrac{1}{x}\dfrac{{dy}}{{dx}} + 4{x^2}\left( {9y + 6} \right) = 0$
can be written as

$\dfrac{{{d^2}y}}{{d{t^2}}} + 9y = - 6$ .

Hence find the general solution.

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