Linear ODE

A linear ODE is one where the unknown function and its derivatives appear only to the first power, with no products between them, and the coefficients depend only on the independent variable. The general form for $n$-th order:

$a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdots +a_1(x)y^{\prime }+a_0(x)y=f(x)$

The coefficients $a_i(x)$ can be constants or functions of $x$. They cannot depend on $y$ or any of its derivatives — that would make the equation nonlinear.

What “linear” means

The ODE is linear in $y$ and its derivatives. The function $f(x)$ on the right-hand side can be anything — it’s the forcing term or inhomogeneity, and the equation is linear regardless of what $f(x)$ looks like.

Examples that are linear:

• $y^{\prime }+2y=e^x$ — first-order, constant coefficients.
• $x^2{y}^{\prime \prime }+x{y}^{\prime }+y=\cos x$ — second-order, variable coefficients.
• $y^{(4)}-3y^{\prime \prime }+y=0$ — fourth-order, constant coefficients, homogeneous.

Examples that are not linear:

• $y^{\prime }=y^2$ — $y$ appears to the second power.
• $y{y}^{\prime }=x$ — product of $y$ and its derivative.
• $y^{\prime \prime }+\sin y=0$ — $y$ inside a transcendental function (the pendulum equation).
• $(y^{\prime }{)}^2+y=x$ — derivative squared.

Homogeneous vs nonhomogeneous

Two flavors based on whether the right-hand side is zero:

• Homogeneous: $f(x)=0$. The equation reduces to $a_n(x)y^{(n)}+\cdots +a_0(x)y=0$.
• Nonhomogeneous: $f(x)\ne 0$. The forcing term drives the solution away from zero.

Homogeneous equations have nicer structure. The Superposition principle applies: any linear combination of solutions is itself a solution. Solutions form a vector space.

For a nonhomogeneous equation, the general solution decomposes as

$y=y_c+y_p$

where $y_c$ is the complementary solution (general solution of the homogeneous part) and $y_p$ is any particular solution of the nonhomogeneous equation. See Particular solution and complementary solution.

Why linearity matters

Linear ODEs admit:

1. Superposition. Solutions add. Any linear combination of homogeneous solutions is a homogeneous solution.
2. Closed-form solutions for many cases. First-order linear is always solvable via Integrating factor. Second-order linear with constant coefficients is solvable via the Characteristic equation.
3. Existence and uniqueness with stronger conclusions. For linear ODEs, the Existence and uniqueness theorem guarantees global solutions on the entire interval where the coefficients are continuous, not just locally. The hypotheses are actually weaker than the nonlinear theorem demands — linear needs only continuity of the coefficients $P(x),Q(x)$, whereas the general nonlinear case requires continuity of $f$ and $\partial f/\partial y$. So linear ODEs ask for less and deliver more: from weaker assumptions you still get a stronger (global, not local) conclusion.
4. Useful transforms. The Laplace transform turns linear ODEs into algebraic equations.

Nonlinear ODEs lose all of these. There’s no general method; each one is solved by its own tricks (or, often, only numerically).

Solution methods for linear ODEs

By order:

First-order linear: $a_1(x)y^{\prime }+a_0(x)y=f(x)$. Use Integrating factor.

Second-order linear, constant coefficients, homogeneous: $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$. Use Characteristic equation.

Second-order linear, constant coefficients, nonhomogeneous: solve the homogeneous part for $y_c$, then find $y_p$ via Method of undetermined coefficients (when $f(x)$ has a nice form) or Method of variation of parameters (general $f$).

Higher-order or variable coefficients: same general approach but more involved. Method of reduction of order and Laplace transform become important.

For nonlinear ODEs, see Separable equation and Exact equation for the limited cases that can be solved directly.