Lesson 12.3 - Another Theory of Linear Differential Equations (Theorem 2)

Let $y_p$ be a particular solution of the nonhomogeneous equation. Consider the $n$th order linear differential equation

$y^{(n)}(x)+p_1(x)y^{(n-1)}(x)+p_2(x)y^{(n-2)}(x)\cdots +p_{(n-1)}{y}^{\prime }(x)+p_{n}y(x)=f(x)$

where $p_1(x),p_2(x),\dots ,p_n(x)$ are functions of $x$ alone on an interval $I=(a,b)$. Let $f_1,f_2,\dots ,f_n$ be a fundamental solution set for the corresponding homogeneous equation

$y^{(n)}(x)+p_1(x)y^{(n-1)}(x)+p_2(x)y^{(n-2)}(x)\cdots +p_{(n-1)}{y}^{\prime }(x)+p_{n}y(x)=0$

Then every solution of the nonhomogeneous equation on $I$ can be written as

$y(x)=y_p+c_1{y}_1+c_2{y}_2+\cdots +c_n{y}_n$

Next Lesson: Lesson 13 - Mass Spring Systems

Table of Contents: Table of Contents