Undetermined coefficients

Problem 6.364 · hard

1. \[ r^{2} + 5 r + 4 = \left(r + 1\right) \left(r + 4\right) \]
   The characteristic equation has roots -4 and -1.✓ Proved
2. So y_h = C₁e^(-4x) + C₂e^(-1x). Guess y_p = A*x + B.
   Reviewed
3. \[ x + 5 \frac{d}{d x} \left(\frac{x}{4} - \frac{21}{16}\right) + \frac{d^{2}}{d x^{2}} \left(\frac{x}{4} - \frac{21}{16}\right) - \frac{21}{4} = x - 4 \]
   Matching coefficients gives y_p = x/4 - 21/16; it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. \frac{x}{4} - \frac{21}{16} + \frac{7 e^{- x}}{3} - \frac{e^{- 4 x}}{48} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(\frac{x}{4} - \frac{21}{16} + \frac{7 e^{- x}}{3} - \frac{e^{- 4 x}}{48}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}1\\-2\end{matrix}\right] \]
   The initial conditions fix C₁ = -1/48 and C₂ = 7/3.✓ Proved