superposition

Principle of Superposition

Statement

Any linear combination of solutions to a homogeneous linear ODE is also a solution.

Assumptions

• The ODE is linear and homogeneous: $\ddot{x}+p(t)\dot{x}+q(t)x=0$.
• $X_1(t)$ and $X_2(t)$ are solutions on a common interval.
• $c_1,c_2$ are arbitrary constants.

Proof Sketch

Because differentiation is linear, substituting $x=c_1{X}_1+c_2{X}_2$ into the ODE separates into two copies of the original equation — each equal to zero — whose sum is zero. No nonlinear terms appear, so the combination is itself a solution.

Full Proof

Let $X_1$ and $X_2$ satisfy the ODE:

$${\ddot{X}}_i+p(t){\dot{X}}_i+q(t)X_i=0,i=1,2.$$

Set $x=c_1{X}_1+c_2{X}_2$. Then

$$\ddot{x}+p\dot{x}+qx=c_1\left({\ddot{X}}_1+p{\dot{X}}_1+q{X}_1\right)+c_2\left({\ddot{X}}_2+p{\dot{X}}_2+q{X}_2\right)=c_1\cdot 0+c_2\cdot 0=0.$$

Hence $x=c_1{X}_1+c_2{X}_2$ satisfies the ODE. $\square$

Notes / Intuition

• Superposition holds for any linear homogeneous ODE; it fails for nonlinear ODEs.
• The set of all solutions to a second-order homogeneous linear ODE forms a two-dimensional vector space; any two linearly independent solutions form a basis.
• Superposition is the foundation for building the general solution $x_h=c_1{X}_1+c_2{X}_2$ used in the inhomogeneous solution method.
• Linear independence of $X_1$ and $X_2$ is verified by the Wronskian: if $W\ne 0$, the two solutions span the full solution space.

Links

• Homogeneous Second-Order ODE
• Wronskian
• Inhomogeneous Second-Order ODE