Undetermined coefficients

Problem 6.407 · hard

1. \[ r^{2} - 1 = \left(r - 1\right) \left(r + 1\right) \]
   The characteristic equation has roots -1 and 1.✓ Proved
2. So y_h = C₁e^(-1x) + C₂e^(1x). Guess y_p = A*cos(3*x) + B*sin(3*x).
   Reviewed
3. \[ - \frac{\cos{\left(3 x \right)}}{2} + \frac{d^{2}}{d x^{2}} \frac{\cos{\left(3 x \right)}}{2} = - 5 \cos{\left(3 x \right)} \]
   Matching coefficients gives y_p = cos(3*x)/2; it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. \frac{e^{x}}{4} + \frac{\cos{\left(3 x \right)}}{2} + \frac{9 e^{- x}}{4} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(\frac{e^{x}}{4} + \frac{\cos{\left(3 x \right)}}{2} + \frac{9 e^{- x}}{4}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}3\\-2\end{matrix}\right] \]
   The initial conditions fix C₁ = 9/4 and C₂ = 1/4.✓ Proved