The particular integral

3 The particular integral

Given a second order ODE

$a \frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = f (x),$

a particular integral is any function, $y_{p} (x)$ , which satisfies the equation. That is, any function which when substituted into the left-hand side, results in the expression on the right-hand side.

Task!

Show that

$y = - \frac{1}{4} e^{2 x}$

is a particular integral of

$\frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} - 6 y = e^{2 x}$ (1)

Starting with $y = - \frac{1}{4} e^{2 x}$ , find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ :

$\frac{d y}{d x} = - \frac{1}{2} e^{2 x}$ , $\frac{d^{2} y}{d x^{2}} = - e^{2 x}$ Now substitute these into the ODE and simplify to check it satisfies the equation:

Substitution yields $- e^{2 x} - (- \frac{1}{2} e^{2 x}) - 6 (- \frac{1}{4} e^{2 x})$ which simplifies to $e^{2 x}$ , the same as the right-hand side.

Therefore $y = - \frac{1}{4} e^{2 x}$ is a particular integral and we write (attaching a subscript p):

$y_{p} (x) = - \frac{1}{4} e^{2 x}$

Task!

State what is meant by a particular integral.

A particular integral is any solution of a differential equation.

3 The particular integral

Task!

Answer

Answer

Task!

Answer