First-order linear equations

Problem 6.178 · hard

1. The equation is linear in standard form; the integrating factor is e^(∫3 dx) = e^(3x).
   Reviewed
2. \[ \frac{d}{d x} Y{\left(x \right)} e^{3 x} = 3 Y{\left(x \right)} e^{3 x} + e^{3 x} \frac{d}{d x} Y{\left(x \right)} \]
   Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
3. \[ \int 3 x e^{3 x}\, dx = \frac{\left(3 x - 1\right) e^{3 x}}{3} \]
   Integrate the right side.✓ Proved
4. Setting x = 0 and y = -1 fixes the constant of integration: C = -2/3.
   Reviewed
5. \[ 3 x + \frac{d}{d x} \left(x - \frac{1}{3} - \frac{2 e^{- 3 x}}{3}\right) - 1 - 2 e^{- 3 x} = 3 x \]
   The solution satisfies the equation.✓ Proved
6. \[ -1 \]
   And the initial condition.✓ Proved