The Superposition Principle

Let $y_1$ be a solution to the DE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=f_1(t)$, and $y_2$ a solution to the DE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=f_2(t)$

Then for any constants $c_1$ and $c_2$, the function $c_1{y}_1+c_2{y}_2$ is a solution to the DE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=c_1{f}_1(t)+c_2{f}_2(t)$

Application

$$y_g=y_h+y_p=c_1{y}_1+c_2{y}_2+y_p$$

• $y_p$ is a particular solution to $a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)$
• $y_h$ is a general solution to $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$