Review / Part II — Higher-Order & Nonlinear / Ch 11

Variation of Parameters

Ch 11 — Part II — Higher-Order & Nonlinear

Variation of parameters constructs a particular solution of a linear ODE from a known basis of the homogeneous equation, for arbitrary continuous forcing \(g(t)\).

Formula (2nd order)

Let \(y_1, y_2\) be a basis of solutions of \(y'' + p y' + q y = 0\), with Wronskian \(W = y_1 y_2' - y_1' y_2\). For \(y'' + p y' + q y = g(t)\), $$y_p(t) = -y_1(t) \int \frac{y_2(t) g(t)}{W(t)}\,dt + y_2(t) \int \frac{y_1(t) g(t)}{W(t)}\,dt.$$

When to use

Use when undetermined coefficients doesn't apply (e.g. \(g(t) = \tan t\), \(1/t\), \(\ln t\)). Works for any continuous forcing, at the cost of computing two integrals.

Practice Problems

Problem 1hard
Find a particular solution to \(y'' + y = \sec t\).
Hint
\(y_1 = \cos t, y_2 = \sin t\), \(W = 1\).
Solution

\(y_p = -\cos t \int \sin t \sec t \,dt + \sin t \int \cos t \sec t \,dt = -\cos t \int \tan t \,dt + \sin t \int 1\,dt = -\cos t(-\ln|\cos t|) + t \sin t = \cos t \ln|\cos t| + t \sin t\).

Answer: \(y_p = \cos t\,\ln|\cos t| + t\sin t.\)

Problem 2medium
For \(y'' - y = e^t\), apply variation of parameters.
Hint
\(y_1 = e^t, y_2 = e^{-t}\), \(W = -2\).
Solution

\(y_p = -e^t \int \frac{e^{-t}\cdot e^t}{-2}\,dt + e^{-t} \int \frac{e^t \cdot e^t}{-2}\,dt = \tfrac{1}{2} e^t \cdot t - \tfrac{1}{4} e^{-t} e^{2t} = \tfrac{t}{2} e^t - \tfrac{1}{4} e^t\). The second term is in the homogeneous kernel and can be absorbed.

Answer: \(y_p = \tfrac{t}{2} e^t\) (up to homogeneous).

Problem 3easy
State the hypothesis needed for the VoP formula to work.
Hint
Need a known basis and continuity.
Solution

\(y_1, y_2\) linearly independent solutions of the homogeneous equation; \(p, q, g\) continuous on the interval.

Answer: See solution.