Undetermined coefficients

Problem 6.409 · hard

1. \[ r^{2} - 4 r + 3 = \left(r - 3\right) \left(r - 1\right) \]
   The characteristic equation has roots 1 and 3.✓ Proved
2. So y_h = C₁e^(1x) + C₂e^(3x). Guess y_p = A*cos(3*x) + B*sin(3*x).
   Reviewed
3. \[ \frac{3 \sin{\left(3 x \right)}}{10} - \frac{3 \cos{\left(3 x \right)}}{5} - 4 \frac{d}{d x} \left(\frac{\sin{\left(3 x \right)}}{10} - \frac{\cos{\left(3 x \right)}}{5}\right) + \frac{d^{2}}{d x^{2}} \left(\frac{\sin{\left(3 x \right)}}{10} - \frac{\cos{\left(3 x \right)}}{5}\right) = - 3 \sin{\left(3 x \right)} \]
   Matching coefficients gives y_p = sin(3*x)/10 - cos(3*x)/5; it satisfies the equation.✓ Proved
4. \[ \left[\begin{matrix}\left. - \frac{e^{3 x}}{4} + \frac{9 e^{x}}{20} + \frac{\sin{\left(3 x \right)}}{10} - \frac{\cos{\left(3 x \right)}}{5} \right|_{\substack{ x=0 }}\\\left. \frac{d}{d x} \left(- \frac{e^{3 x}}{4} + \frac{9 e^{x}}{20} + \frac{\sin{\left(3 x \right)}}{10} - \frac{\cos{\left(3 x \right)}}{5}\right) \right|_{\substack{ x=0 }}\end{matrix}\right] = \left[\begin{matrix}0\\0\end{matrix}\right] \]
   The initial conditions fix C₁ = 9/20 and C₂ = -1/4.✓ Proved