First-order linear equations

Problem 6.126 · 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. \[ \int x e^{2 x}\, dx = \frac{\left(2 x - 1\right) e^{2 x}}{4} \]
   Integrate the right side.✓ Proved
4. Setting x = 0 and y = 4 fixes the constant of integration: C = 17/4.
   Reviewed
5. \[ x + \frac{d}{d x} \left(\frac{x}{2} - \frac{1}{4} + \frac{17 e^{- 2 x}}{4}\right) - \frac{1}{2} + \frac{17 e^{- 2 x}}{2} = x \]
   The solution satisfies the equation.✓ Proved
6. \[ 4 \]
   And the initial condition.✓ Proved