Undetermined coefficients

Problem 6.288 · hard

1. \[ r^{2} + 2 r - 3 = \left(r - 1\right) \left(r + 3\right) \]
   The characteristic equation has roots 1 and -3.✓ Proved
2. So y_h = C₁e^(1x) + C₂e^(-3x). Guess y_p = A.
   Reviewed
3. \[ 2 \frac{d}{d x} 3 + \frac{d^{2}}{d x^{2}} 3 - 9 = -9 \]
   Matching coefficients gives y_p = 3; it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. - \frac{17 e^{x}}{4} + 3 - \frac{7 e^{- 3 x}}{4} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- \frac{17 e^{x}}{4} + 3 - \frac{7 e^{- 3 x}}{4}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}-3\\1\end{matrix}\right] \]
   The initial conditions fix C₁ = -17/4 and C₂ = -7/4.✓ Proved