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

Undetermined Coefficients (2nd order)

Ch 11 — Part II — Higher-Order & Nonlinear

For \(y'' + a y' + b y = g(t)\) where \(g\) is a polynomial, exponential, sine/cosine, or product/sum of these, guess a particular solution with the same form as \(g\) and solve for coefficients.

Guess templates

\(g(t)\)Guess \(y_p\)
polynomial of degree \(n\)\(A_n t^n + \cdots + A_1 t + A_0\)
\(e^{\alpha t}\)\(A e^{\alpha t}\)
\(\cos(\omega t)\) or \(\sin(\omega t)\)\(A\cos\omega t + B\sin\omega t\)
product of the aboveproduct of the above guesses

Resonance fix

If the guess already solves the homogeneous equation, multiply by the smallest power of \(t\) that breaks the overlap. Example: \(y'' - y = e^t\) — the homogeneous kernel contains \(e^t\), so guess \(y_p = A t e^t\).

Full solution

\(y = y_h + y_p\); fit initial conditions at the end.

Practice Problems

Problem 1medium
Find a particular solution to \(y'' + y = 3 t^2\).
Hint
Polynomial guess \(y_p = A t^2 + B t + C\).
Solution

\(y_p' = 2At + B,\; y_p'' = 2A\). Plug in: \(2A + At^2 + Bt + C = 3t^2\Rightarrow A = 3,\; B = 0,\; 2A + C = 0\Rightarrow C = -6\).

Answer: \(y_p = 3t^2 - 6.\)

Problem 2hard
Solve \(y'' + 4y = \cos(2t)\) (resonance).
Hint
Cosine matches homogeneous freq ⇒ multiply by \(t\).
Solution

Homogeneous: \(\cos 2t, \sin 2t\). Guess \(y_p = t(A\cos 2t + B\sin 2t)\). Compute \(y_p'' + 4y_p\)... result: \(-4A\sin 2t + 4B\cos 2t = \cos 2t\Rightarrow B = 1/4,\; A = 0\).

Answer: \(y_p = \tfrac{t}{4}\sin(2t).\)

Problem 3medium
Find a particular solution to \(y'' - 3y' + 2y = e^{3t}\).
Hint
Guess \(A e^{3t}\).
Solution

\((9 - 9 + 2) A e^{3t} = e^{3t}\Rightarrow 2A = 1\Rightarrow A = 1/2\).

Answer: \(y_p = \tfrac{1}{2} e^{3t}.\)