Higher-Order Linear ODEs

Linear Equations of Order n

$$L[y] = y^{(n)} + p_{n-1}(t)y^{(n-1)} + \cdots + p_1(t)y' + p_0(t)y = g(t)$$

• If $p_0, \ldots, p_{n-1}, g$ are continuous on an interval containing $t_0$, the initial value problem has a unique solution there.
• For the homogeneous equation $L[y] = 0$, if $y_1, y_2$ are solutions so is $c_1 y_1 + c_2 y_2$.
• The general solution of $L[y] = g$ is $y = y_h + Y$, the general homogeneous solution plus any particular solution.

Wronskian

$$W[y_1, \ldots, y_n](t) = \det \begin{bmatrix} y_1 & y_2 & \cdots & y_n \\ y_1' & y_2' & \cdots & y_n' \\ \vdots & & & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)} \end{bmatrix}$$

For solutions of $L[y] = 0$, $W \neq 0$ at one point iff $y_1, \ldots, y_n$ form a fundamental set (linearly independent).

Constant Coefficients

Substituting $e^{rt}$ gives the characteristic polynomial. Each root contributes

• a real root $r$: $e^{rt}$,
• a root $r$ of multiplicity $k$: $e^{rt}, t e^{rt}, \ldots, t^{k-1}e^{rt}$,
• a complex pair $\lambda \pm i\mu$: $e^{\lambda t}\cos(\mu t)$ and $e^{\lambda t}\sin(\mu t)$.

Reduction of Order

Given one solution $y_1$ of $y'' + p(t)y' + q(t)y = 0$, set $y_2 = v(t) y_1$ and solve for $v$, which gives

$$\frac{y_2}{y_1} = \int \frac{e^{-\int p(t)\, dt}}{y_1^2}\, dt$$

Variation of Parameters

For $y'' + p(t)y' + q(t)y = g(t)$ with fundamental solutions $y_1, y_2$,

$$Y(t) = -y_1(t) \int_{t_0}^{t} \frac{y_2(s) g(s)}{W[y_1, y_2](s)}\, ds + y_2(t) \int_{t_0}^{t} \frac{y_1(s) g(s)}{W[y_1, y_2](s)}\, ds$$

For order $n$, look for $Y = v_1 y_1 + \cdots + v_n y_n$ with

$$\begin{cases} v_1' y_1^{(k)} + \cdots + v_n' y_n^{(k)} = 0, & k = 0, \ldots, n-2 \\ v_1' y_1^{(n-1)} + \cdots + v_n' y_n^{(n-1)} = g \end{cases}$$

solve this linear system for the $v_i'$ (its determinant is the Wronskian), and integrate.

Undetermined Coefficients

For constant coefficients and $g$ of a special form, guess $Y$ of the same form,

• $P_n(t) \to A_n t^n + \cdots + A_1 t + A_0$,
• $e^{at} \to A e^{at}$,
• $\sin(\beta t)$ or $\cos(\beta t) \to A\cos(\beta t) + B\sin(\beta t)$.

If a term of the guess solves the homogeneous equation, multiply the guess by $t^s$, where $s$ is the multiplicity of the corresponding characteristic root.