Theory for Inhomogeneous ODE's

June 9, 2019

General Solution

If an n-th order linear non-homogeneous equations is in this form: $y^{(n)}+p_1y^{(n-1)}+...+p_{n-1}y=f(t)\tag{1}$ then its associated homogeneous equation is: $y^{(n)}+p_1y^{(n-1)}+...+p_{n-1}y=0$ We know that there exists $n$ independent solutions for an n-th order homogeneous DE: $y_c=c_1y_1+...+c_ny_n$ $y_c$ here is called complementary solution. Suppose there’s a $y_p$ that can solve (1), (1)’s general solution is $Y=y_c+y_p$ Proof: substitute $Y$ into (1): $\begin{align} Y^{(n)}+p_1Y^{(n-1)}+...+p_{n-1}Y&=f(t)\\ y_c^{(n)}+y_p^{(n)}+p_1(y_c^{(n-1)}+y_p^{(n-1)})+...+p_{n-1}(y_c+y_p)&=f(t)\\ y_c^{(n)}+p_1y_c^{(n-1)}+...+p_{n-1}y_c+(...)&=f(t)\\ 0+(...)&=f(t) \end{align}$ thus adding $y_c$ does not affect the equation and $Y$ is the general solution.