Ordinary differential equation

An ordinary differential equation (ODE) is a Differential equation involving derivatives with respect to a single independent variable. The “ordinary” distinguishes it from a partial differential equation (PDE), which involves multiple independent variables and partial derivatives.

The general form:

$F(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)})=0$

where $y=y(x)$ is the unknown function and $y^{(k)}$ is its $k$-th derivative.

Order

The order of an ODE is the order of its highest derivative.

• $y^{\prime }=3x^2$ — first-order.
• $y^{\prime \prime }+2y^{\prime }+y=0$ — second-order.
• $y^{(4)}+y=\sin x$ — fourth-order.

Higher-order ODEs need more side conditions to specify a unique solution. A first-order ODE with $y(x_0)=y_0$ pins down a unique solution; a second-order needs both $y(x_0)$ and $y^{\prime }(x_0)$.

Examples by application

• $\frac{dy}{dx}=ky$ — exponential growth/decay (population, radioactive decay).
• $m{y}^{\prime \prime }+\gamma y^{\prime }+ky=0$ — damped mass-spring oscillator. See Mechanical and electrical vibrations.
• $L\frac{di}{dt}+Ri=V(t)$ — RL circuit with applied voltage.

Each is an ODE because the unknown function depends on a single variable (time, position, etc.).

Linear vs nonlinear

An ODE is linear if every term involving $y$ or its derivatives is first-degree linear in those quantities together: each such term is a (function of $x$) times a single one of $y,y^{\prime },y^{\prime \prime },\dots$. Equivalently:

1. $y,y^{\prime },y^{\prime \prime },\dots$ appear only to the first power. (This rule alone implies the second.)
2. No products like $y\cdot y^{\prime }$ — these are quadratic in the $(y,y^{\prime },\dots )$ tuple even though each factor appears only to the first power. The condition is “first-degree as a polynomial in $(y,y^{\prime },\dots )$ jointly,” which forbids products as well as squares.
3. Coefficients can depend on $x$ but not on $y$.

A linear ODE of order $n$ has the form:

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

Linear ODEs have well-developed solution theory, the Superposition principle, and predictable behavior. Nonlinear ODEs (like $y^{\prime }=y^2+x$) are harder — many can’t be solved in closed form.

Confirming a solution

A function $\varphi (x)$ is a solution of an $n$-th order ODE on some interval $I=(a,b)$ if:

1. $\varphi$ is $n$-times differentiable on $I$.
2. The equation $F(x,\varphi ,{\varphi }^{\prime },\dots ,{\varphi }^{(n)})=0$ holds for every $x\in I$.

Example: is $\varphi (x)=x^2-\frac{1}{x}$ a solution of $y^{\prime \prime }-\frac{2}{x^2}y=0$ on $x>0$?

Compute: ${\varphi }^{\prime }=2x+\frac{1}{x^2}$, ${\varphi }^{\prime \prime }=2-\frac{2}{x^3}$.

Substitute: ${\varphi }^{\prime \prime }-\frac{2}{x^2}\varphi =\left(2-\frac{2}{x^3}\right)-\frac{2}{x^2}\left(x^2-\frac{1}{x}\right)=2-\frac{2}{x^3}-2+\frac{2}{x^3}=0$.

So yes — $\varphi$ satisfies the ODE on $x>0$.

The general principle: confirming a candidate solution means taking the derivatives and plugging in. Don’t trust a printed claim — check.

Solving ODEs

Different ODE types have different solution methods. The systematic ones, in roughly increasing order of complexity:

• Separable equation — first-order, separates into pure-$x$ and pure-$y$ parts.
• Integrating factor — first-order linear with variable coefficients.
• Exact equation — first-order, when a potential function exists.
• Picard iteration — first-order, iterative approximation.
• Characteristic equation — second-order linear with constant coefficients.
• Method of undetermined coefficients — nonhomogeneous, when $f(x)$ has a nice form.
• Method of variation of parameters — nonhomogeneous, general $f(x)$.
• Method of reduction of order — second-order when one solution is known.
• Laplace transform — converts ODEs to algebraic equations.

For systems of ODEs (multiple unknowns), see System of first-order linear ODEs.

For when a solution is guaranteed to exist, see Existence and uniqueness theorem.