Undetermined coefficients

Problem 6.284 · hard

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