First-order linear equations

Problem 6.123 · hard

1. The equation is linear in standard form; the integrating factor is e^(∫2 dx) = e^(2x).
   Reviewed
2. \[ \frac{d}{d x} Y{\left(x \right)} e^{2 x} = 2 Y{\left(x \right)} e^{2 x} + e^{2 x} \frac{d}{d x} Y{\left(x \right)} \]
   Multiplying by e^(ax) turns the left side into (e^(ax) y)'.✓ Proved
3. \[ \int 2 e^{3 x}\, dx = \frac{2 e^{3 x}}{3} \]
   Integrate the right side.✓ Proved
4. Setting x = 0 and y = 3 fixes the constant of integration: C = 7/3.
   Reviewed
5. \[ \frac{\left(4 e^{3 x} + 14\right) e^{- 2 x}}{3} + \frac{d}{d x} \frac{\left(2 e^{3 x} + 7\right) e^{- 2 x}}{3} = 2 e^{x} \]
   The solution satisfies the equation.✓ Proved
6. \[ 3 \]
   And the initial condition.✓ Proved