Undetermined coefficients

Problem 6.287 · hard

1. \[ r^{2} - 5 r + 6 = \left(r - 3\right) \left(r - 2\right) \]
   The characteristic equation has roots 2 and 3.✓ Proved
2. So y_h = C₁e^(2x) + C₂e^(3x). Guess y_p = A*exp(x) (the exponent is not a root, so no extra factor of x is needed).
   Reviewed
3. \[ - 12 e^{x} - 5 \frac{d}{d x} \left(- 2 e^{x}\right) + \frac{d^{2}}{d x^{2}} \left(- 2 e^{x}\right) = - 4 e^{x} \]
   Matching coefficients gives y_p = -2*exp(x); it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. - 7 e^{3 x} + 11 e^{2 x} - 2 e^{x} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- 7 e^{3 x} + 11 e^{2 x} - 2 e^{x}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}2\\-1\end{matrix}\right] \]
   The initial conditions fix C₁ = 11 and C₂ = -7.✓ Proved