Undetermined coefficients

Problem 6.286 · hard

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