First-order linear equations

Problem 6.153 · hard

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