Superposition Principle

Definition

If $y_1,y_2$ are solutions of homogeneous linear ODE $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=r(x)$ on $I=(\alpha ,\beta )$ then, any linear combination $c_1{y}_1+c_2{y}_2(c_1,c_2\in \mathbb{R})$ is also a solution (on $I$)

Facts

Suppose that each of infinitely many functions $u_1,u_2,\dots$ satisfies a linear homogeneous differential equation or boundary condition $L(u)=0$

Then, the infinite series $u={\sum }_{n=1}^{\infty }{c}_n{u}_n$ where $c_n\in \mathbb{R}\text{or}\mathbb{C}$ that satisfies following conditions satisfies $L(u)=0$

• the infinite series converges and required differentiability in $L(u)=0$
• the required boundary condition is satisfied