Second-order linear equations

Problem 6.134 · hard

1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
   Reviewed
2. \[ \left(m + 1 - 3 i\right) \left(m + 1 + 3 i\right) = m^{2} + 2 m + 10 \]
   Roots: -1 - 3*I, -1 + 3*I.✓ Proved
3. Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
   Reviewed
4. Use y(0) and y'(0) to find C₁ and C₂.
   Reviewed
5. \[ \frac{\left(- 20 \sin{\left(3 x \right)} - 90 \cos{\left(3 x \right)}\right) e^{- x}}{3} + 2 \frac{d}{d x} \frac{\left(- 2 \sin{\left(3 x \right)} - 9 \cos{\left(3 x \right)}\right) e^{- x}}{3} + \frac{d^{2}}{d x^{2}} \frac{\left(- 2 \sin{\left(3 x \right)} - 9 \cos{\left(3 x \right)}\right) e^{- x}}{3} = 0 \]
   The solution satisfies the equation.✓ Proved
6. \[ -3 \]
   y(0).✓ Proved
7. \[ 1 \]
   y'(0).✓ Proved