First-order linear equations

Problem 6.129 · medium

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. \[ \int 6 e^{x}\, dx = 6 e^{x} \]
   Integrate the right side.✓ Proved
4. Setting x = 0 and y = 3 fixes the constant of integration: C = -3.
   Reviewed
5. \[ \frac{d}{d x} \left(6 - 3 e^{- x}\right) + 6 - 3 e^{- x} = 6 \]
   The solution satisfies the equation.✓ Proved
6. \[ 3 \]
   And the initial condition.✓ Proved