First-order linear equations

Problem 6.104 · hard

1. The equation is linear in standard form; the integrating factor is e^(∫4 dx) = e^(4x).
   Reviewed
2. \[ \frac{d}{d x} Y{\left(x \right)} e^{4 x} = 4 Y{\left(x \right)} e^{4 x} + e^{4 x} \frac{d}{d x} Y{\left(x \right)} \]
   Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
3. \[ \int 3 x e^{4 x}\, dx = \frac{\left(12 x - 3\right) e^{4 x}}{16} \]
   Integrate the right side.✓ Proved
4. Setting x = 0 and y = -1 fixes the constant of integration: C = -13/16.
   Reviewed
5. \[ 3 x + \frac{d}{d x} \left(\frac{3 x}{4} - \frac{3}{16} - \frac{13 e^{- 4 x}}{16}\right) - \frac{3}{4} - \frac{13 e^{- 4 x}}{4} = 3 x \]
   The solution satisfies the equation.✓ Proved
6. \[ -1 \]
   And the initial condition.✓ Proved