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.

  1. 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.

2nd Order Differential. Equations - Making a substitution : ExamSolutions Maths Revision - youtube Video