1 Constant coefficient second order linear ODEs

We now proceed to study those second order linear equations which have constant coefficients. The general form of such an equation is:

$a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=f\left(x\right)$ (3)

where $a,b,c$ are constants. The homogeneous form of (3) is the case when $f\left(x\right)\equiv 0$ :

$a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=0$ (4)

To find the general solution of (3), it is first necessary to solve (4). The general solution of (4) is called the complementary function and will always contain two arbitrary constants. We will denote this solution by $y_{\text{cf}}$ .

The technique for finding the complementary function is described in this Section.

Task!

State which of the following are constant coefficient equations.

State which are homogeneous.

1. $\frac{d^{2}y}{d{x}^2}+4\frac{dy}{dx}+3y={\text{e}}^{-2x}$
2. $x\frac{d^{2}y}{d{x}^2}+2y=0$
3. $\frac{d^{2}x}{d{t}^2}+3\frac{dx}{dt}+7x=0$   
4. $\frac{d^{2}y}{d{x}^2}+4\frac{dy}{dx}+4y=0$

Answer

Note: A complementary function is the general solution of a homogeneous, linear differential equation.