Superposition principle

The superposition principle states that for a homogeneous linear ODE, any linear combination of solutions is itself a solution. If ${\varphi }_1,{\varphi }_2,\dots ,{\varphi }_k$ are all solutions of

$a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdots +a_0(x)y=0$

then for any constants $c_1,c_2,\dots ,c_k$,

$\varphi (x)=c_1{\varphi }_1(x)+c_2{\varphi }_2(x)+\cdots +c_k{\varphi }_k(x)$

is also a solution. The solution space is closed under linear combinations — it’s a vector space.

This is the foundation of how we solve linear ODEs: find a few specific solutions, then build the general solution as their linear combination.

Why it works

Linearity. Every term in the ODE is linear in $y$ and its derivatives. So if you plug in $\varphi =c_1{\varphi }_1+c_2{\varphi }_2$:

$a_n(x)(c_1{\varphi }_1+c_2{\varphi }_2{)}^{(n)}=c_1{a}_n(x){\varphi }_1^{(n)}+c_2{a}_n(x){\varphi }_2^{(n)}$

Each term distributes over the linear combination. Sum every term in the ODE:

${\sum }_k{a}_k(x)(c_1{\varphi }_1+c_2{\varphi }_2{)}^{(k)}=c_1{\sum }_k{a}_k(x){\varphi }_1^{(k)}+c_2{\sum }_k{a}_k(x){\varphi }_2^{(k)}=c_1\cdot 0+c_2\cdot 0=0$

Each individual sum is zero because ${\varphi }_1$ and ${\varphi }_2$ are solutions. So $\varphi =c_1{\varphi }_1+c_2{\varphi }_2$ also makes the equation equal zero.

Where the principle fails

Superposition holds only for homogeneous linear ODEs. It fails:

• For nonhomogeneous linear ODEs. If ${\varphi }_1,{\varphi }_2$ both satisfy $Ly=f(x)$, then ${\varphi }_1+{\varphi }_2$ satisfies $Ly=2f(x)$, not the original. To handle nonhomogeneous equations, you decompose the solution as $y_c+y_p$ — see Particular solution and complementary solution.
• For nonlinear ODEs. If ${\varphi }_1,{\varphi }_2$ both satisfy $y^{\prime }=y^2$, then ${\varphi }_1+{\varphi }_2$ usually doesn’t. Linear combinations of solutions are not solutions.

So superposition is uniquely a property of homogeneous linear equations — but those happen to be a major class with deep theory.

Practical use

In second-order homogeneous linear ODEs (like $y^{\prime \prime }+2y^{\prime }+y=0$), you find two linearly independent solutions $y_1,y_2$ via the Characteristic equation, then write:

$y(x)=c_1{y}_1(x)+c_2{y}_2(x)$

This is the general solution. The two arbitrary constants are pinned down by initial conditions. Without superposition, you’d have to find one specific solution per initial condition pair — exhausting and unsystematic.

For systems of linear ODEs, superposition extends naturally — see Linear independence of vector functions and System of first-order linear ODEs.

For the linear-independence requirement that ensures you have enough solutions to span the solution space, see Wronskian and Representation theorem.

Superposition in circuits (the EE form)

This note covers the ODE form, but the term “superposition” in EE is more often used about circuits with multiple sources: in any linear circuit, the response to several independent sources equals the sum of the responses to each source acting alone (with all other sources “deactivated” — voltage sources replaced by shorts, current sources replaced by opens). It’s the same underlying linearity principle, but applied to the linear network’s input-output map rather than to an ODE’s solution space. EE textbooks call this “the superposition theorem” or “circuit superposition,” and it’s the more common usage in undergraduate electrical engineering.

Superposition for LTI systems

In signals-and-systems, superposition is the operational consequence of linearity and time-invariance combined (LTI): an input can be decomposed into simple pieces (impulses, sinusoids, etc.), the system’s response to each piece is computed independently, and the responses are summed to give the response to the whole input.

This is what justifies the entire convolution-integral machinery: an arbitrary input is decomposed as a continuous sum of weighted impulses, each impulse produces an impulse response, and superposition sums them into the output. It also justifies Fourier series and Fourier-transform analysis: decompose the input into sinusoids, apply $H(j\omega )$ to each, sum.

Without superposition (i.e. without linearity), none of these decompose-respond-sum techniques work. That’s why so much of signal processing is restricted to linear systems and why nonlinear problems usually require linearization around an operating point.