Undetermined coefficients

Problem 6.244 · 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*cos(2*x) + B*sin(2*x).
   Reviewed
3. \[ - \frac{8 \sin{\left(2 x \right)}}{25} - \frac{6 \cos{\left(2 x \right)}}{25} + 3 \frac{d}{d x} \left(\frac{2 \sin{\left(2 x \right)}}{25} + \frac{3 \cos{\left(2 x \right)}}{50}\right) + \frac{d^{2}}{d x^{2}} \left(\frac{2 \sin{\left(2 x \right)}}{25} + \frac{3 \cos{\left(2 x \right)}}{50}\right) = - \sin{\left(2 x \right)} \]
   Matching coefficients gives y_p = 2*sin(2*x)/25 + 3*cos(2*x)/50; it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. \frac{33 e^{x}}{25} + \frac{2 \sin{\left(2 x \right)}}{25} + \frac{3 \cos{\left(2 x \right)}}{50} + \frac{31 e^{- 4 x}}{50} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(\frac{33 e^{x}}{25} + \frac{2 \sin{\left(2 x \right)}}{25} + \frac{3 \cos{\left(2 x \right)}}{50} + \frac{31 e^{- 4 x}}{50}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}2\\-1\end{matrix}\right] \]
   The initial conditions fix C₁ = 33/25 and C₂ = 31/50.✓ Proved