Undetermined coefficients

Problem 6.403 · hard

1. \[ r^{2} + 2 r - 3 = \left(r - 1\right) \left(r + 3\right) \]
   The characteristic equation has roots -3 and 1.✓ Proved
2. So y_h = C₁e^(-3x) + C₂e^(1x). Guess y_p = A*exp(2*x) (the exponent is not a root, so no extra factor of x is needed).
   Reviewed
3. \[ \frac{3 e^{2 x}}{5} + 2 \frac{d}{d x} \left(- \frac{e^{2 x}}{5}\right) + \frac{d^{2}}{d x^{2}} \left(- \frac{e^{2 x}}{5}\right) = - e^{2 x} \]
   Matching coefficients gives y_p = -exp(2*x)/5; it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. - \frac{e^{2 x}}{5} + \frac{11 e^{x}}{4} + \frac{9 e^{- 3 x}}{20} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- \frac{e^{2 x}}{5} + \frac{11 e^{x}}{4} + \frac{9 e^{- 3 x}}{20}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}3\\1\end{matrix}\right] \]
   The initial conditions fix C₁ = 9/20 and C₂ = 11/4.✓ Proved