Superposition Principle

The superposition principle is a fundamental property of linear differential equations that allows us to build complex solutions from simpler ones. This principle is crucial for understanding the structure of solution spaces.

TheoremSuperposition Principle for Homogeneous Equations

If $y_1(x)$ and $y_2(x)$ are solutions of the homogeneous linear differential equation:

$ay'' + by' + cy = 0$

then any linear combination:

$y = c_1y_1(x) + c_2y_2(x)$

is also a solution, where $c_1$ and $c_2$ are arbitrary constants.

Proof

Let $L[y] = ay'' + by' + cy$ be the differential operator. We need to show $L[c_1y_1 + c_2y_2] = 0$ .

By linearity of differentiation:

$L[c_1y_1 + c_2y_2] = a(c_1y_1 + c_2y_2)'' + b(c_1y_1 + c_2y_2)' + c(c_1y_1 + c_2y_2)$

$= c_1(ay_1'' + by_1' + cy_1) + c_2(ay_2'' + by_2' + cy_2)$

$= c_1 \cdot 0 + c_2 \cdot 0 = 0$

since $y_1$ and $y_2$ are solutions. Therefore $c_1y_1 + c_2y_2$ is also a solution.

TheoremGeneral Solution Structure

The general solution of a second-order homogeneous linear ODE is:

$y = c_1y_1 + c_2y_2$

where $y_1$ and $y_2$ are linearly independent solutions forming a fundamental set.

Two solutions are linearly independent if and only if their Wronskian is nonzero:

$W(y_1, y_2) = y_1y_2' - y_1'y_2 \neq 0$

ExampleVerifying Linear Independence

Consider $y_1 = e^{2x}$ and $y_2 = e^{3x}$ , solutions of $y'' - 5y' + 6y = 0$ .

Compute the Wronskian:

$W = e^{2x} \cdot 3e^{3x} - 2e^{2x} \cdot e^{3x} = 3e^{5x} - 2e^{5x} = e^{5x} \neq 0$

Since $W \neq 0$ , the solutions are linearly independent, and:

$y = c_1e^{2x} + c_2e^{3x}$

is the general solution containing all possible solutions.

TheoremSuperposition for Nonhomogeneous Equations

If $y_p$ is a particular solution of the nonhomogeneous equation:

$ay'' + by' + cy = f(x)$

and $y_h = c_1y_1 + c_2y_2$ is the general solution of the associated homogeneous equation, then the general solution of the nonhomogeneous equation is:

$y = y_h + y_p = c_1y_1 + c_2y_2 + y_p$

Remark

The superposition principle fails for nonlinear equations. For example, if $y_1$ and $y_2$ satisfy $y'' + y^2 = 0$ , then $y_1 + y_2$ does NOT satisfy the equation because:

$(y_1+y_2)^2 = y_1^2 + 2y_1y_2 + y_2^2 \neq y_1^2 + y_2^2$

This is why linear equations are fundamentally easier to solve than nonlinear ones.

ExampleMultiple Forcing Functions

Suppose we know particular solutions for:

$y'' + y = \sin x$ has $y_{p1} = -\frac{1}{2}x\cos x$
$y'' + y = \cos x$ has $y_{p2} = \frac{1}{2}x\sin x$

Then by superposition, a particular solution of:

$y'' + y = 3\sin x + 2\cos x$

is:

$y_p = 3y_{p1} + 2y_{p2} = -\frac{3}{2}x\cos x + x\sin x$

The superposition principle is not just a computational tool but reveals the vector space structure of linear differential equations. The solution space of an $n$ -th order homogeneous linear ODE is an $n$ -dimensional vector space.