Superposition Principle for Linear ODE

The superposition principle

The superposition principle is the single most useful property of linear ODEs. It says:

If $y_1(x)$ and $y_2(x)$ are both solutions of a linear homogeneous ODE $L y = 0$, then so is every linear combination $c_1 y_1 + c_2 y_2$, for any constants $c_1, c_2$.

It's the reason the general solution is built as a sum of basis solutions. It's why Fourier series make sense. It's why linear circuits decompose into modes. Almost every computational technique for linear ODEs — characteristic equation, Laplace transform, variation of parameters — leans on this principle to assemble the final answer from simpler pieces.

Why it works — linearity unwrapped

A linear operator is one that respects scalar multiplication and addition: $L(c_1 y_1 + c_2 y_2) = c_1 \, L y_1 + c_2 \, L y_2.$

For the operator $L y = y'' + y$ (and more generally any $L = \sum a_k(x) D^{k}$ with $D = d/dx$), this linearity is a direct consequence of the linearity of differentiation: $(c_1 y_1 + c_2 y_2)' = c_1 y_1' + c_2 y_2'$, and similarly for higher derivatives.

If both $y_1$ and $y_2$ satisfy $L y = 0$, then $L(c_1 y_1 + c_2 y_2) = c_1 \cdot 0 + c_2 \cdot 0 = 0,$ so the linear combination is also a solution.

Critical: this is for homogeneous equations $L y = 0$. For non-homogeneous $L y = g$, superposition holds in a slightly different form (see below).

The given problem

$y'' + y = 0$

Basis: $y_1 = \cos x$, $y_2 = \sin x$ (both solve the equation; see #184 for the characteristic-equation derivation).

Step-by-step verification

Consider $y = c_1 \cos x + c_2 \sin x$.

$y' = -c_1 \sin x + c_2 \cos x$ $y'' = -c_1 \cos x - c_2 \sin x = -y$

Therefore $y'' + y = -y + y = 0. \; \checkmark$

Linear combination works for all $c_1, c_2 \in \mathbb{R}$. $\boxed{}$

Applications — why superposition rules linear theory

1. General solution = linear combination of a basis. For a linear homogeneous $n$-th order ODE, the solution space is $n$-dimensional (see #195 — the Wronskian is nonzero for a linearly independent basis). Every solution is $y = \sum_{i=1}^{n} c_i y_i$ for some basis $\{y_1, \ldots, y_n\}$. Without superposition, the solution space wouldn't be a vector space.

2. Non-homogeneous = particular + homogeneous. For $L y = g(x)$, if $y_p$ is any particular solution and $y_h$ solves $L y_h = 0$, then $y_p + y_h$ solves $L(y_p + y_h) = L y_p + L y_h = g + 0 = g$. The general solution is $y_p + (\text{all } y_h)$.

3. Multiple forcing terms. For $L y = g_1 + g_2$, find $y_{p,1}$ with $L y_{p,1} = g_1$ and $y_{p,2}$ with $L y_{p,2} = g_2$; then $y_p = y_{p,1} + y_{p,2}$ solves the sum. This is the principle of superposition of forces in physics (linear response to multiple sources).

4. Fourier / spectral methods. Decompose a complicated forcing into a sum of simple waves (sines/cosines, exponentials, eigenfunctions), solve for each piece, sum. Only possible because the linear operator distributes over sums.

5. Linear circuits and filters. The response of a linear circuit to a sum of inputs is the sum of responses. That's why you can break a complicated signal into frequency components, filter each, and recombine.

What breaks for non-linear ODEs

Consider $y' = y^{2}$ (a non-linear first-order ODE). If $y_1$ and $y_2$ both solve it, their sum $y_1 + y_2$ probably doesn't: $(y_1 + y_2)' = y_1' + y_2' = y_1^{2} + y_2^{2}$ but we'd need $(y_1 + y_2)' = (y_1 + y_2)^{2} = y_1^{2} + 2 y_1 y_2 + y_2^{2}$. Missing the cross term $2 y_1 y_2$.

Non-linear ODEs have no general superposition. This is why they're so much harder — you can't build solutions from simpler pieces in a general way.

A more complete statement (general linearity)

For the linear operator $L = \sum_{k=0}^{n} a_k(x) D^{k}$ acting on scalar-valued functions, the following are equivalent:

1. $L$ is linear: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2$ for all smooth $y_1, y_2$ and scalars $c_1, c_2$.
2. The set of solutions of $L y = 0$ is a vector space (closed under linear combinations).
3. The solution set of $L y = g$ is an affine subspace $\{y_p + y_h : y_h \in \text{ker}(L)\}$ — a shift of the vector space of homogeneous solutions.

All three statements are the formal expression of superposition at different levels of abstraction.

Wronskian recap

For two solutions $y_1, y_2$ of a linear homogeneous 2nd-order ODE, Wronskian $W(y_1, y_2) \ne 0$ at a point ⇒ they are linearly independent ⇒ they form a basis of the solution space ⇒ every solution is a superposition $c_1 y_1 + c_2 y_2$ (see #195).

Cautionary example — superposition needs both homogeneous ingredients

Let $L y = y'' + y$. Suppose $y_1 = \cos x + 1$ and $y_2 = \sin x + 2$. Do these satisfy $L y = 0$?

$L y_1 = -\cos x + (\cos x + 1) = 1 \ne 0$. So $y_1$ is not a homogeneous solution (it's a non-homogeneous one for $L y = 1$). Superposition won't give a valid homogeneous combo from these.

Moral: before applying superposition, verify each ingredient solves $L y = 0$.

Initial-value problem — assembling the solution

For $y'' + y = 0$ with $y(0) = A$, $y'(0) = B$: $y = C_1 \cos x + C_2 \sin x$ $y(0) = C_1 = A, \quad y'(0) = C_2 = B$ $y(x) = A \cos x + B \sin x.$

Two pieces of initial data pick out one member of the 2-parameter family.

Equivalent amplitude–phase form: $y(x) = R \cos(x - \varphi), \quad R = \sqrt{A^{2} + B^{2}}, \; \varphi = \arctan(B/A).$

The sum of two sinusoids with the same frequency is another sinusoid — nothing new is added by the superposition beyond the 2D space spanned by $\cos, \sin$.

Common mistakes

• Forgetting that superposition fails for non-linear ODEs. A common bug is applying it to "Riccati" or "Bernoulli" equations without first linearising.
• Superposing non-homogeneous solutions. Two solutions of $L y = g$ don't add to give another solution of $L y = g$; they give $L(y_1 + y_2) = 2 g$. The difference of two solutions of $L y = g$ is homogeneous.
• Using linear combinations across different ODEs. Superposition is within one linear operator's solution space — combining solutions of different ODEs gives nothing useful.
• Mistaking the principle for additivity of initial conditions. Be careful: the IVP $L y = 0$, $y(0) = A$, $y'(0) = B$ has a unique solution; superposition is about adding solutions, not about adding initial conditions.

Try it in the visualization

Slide $c_1$ and $c_2$ independently and watch $y = c_1 \cos x + c_2 \sin x$ shift its amplitude and phase continuously — the 2D space of solutions made visible. Toggle to see the amplitude–phase envelope $R = \sqrt{c_1^{2} + c_2^{2}}$ live-update.