A first-order linear DE has the form dy/dx + P(x)y = Q(x). Solve with the integrating factor e^(∫P dx).
Walk through
Step 1 of 5
Put it in standard form
Get the equation to look like dy/dx + P(x)·y = Q(x). Example: dy/dx + 2y = eˣ already fits, with P(x) = 2 and Q(x) = eˣ.
Multiplying by μ = e^(∫P dx) is engineered so the left-hand side collapses into d/dx[μ·y] by the product rule — that's the only reason the method works.
Where first-order linear DEs show up
- Newton's law of cooling — dT/dt + kT = kT_room.
- RC and RL circuits — current with a driving voltage.
- Mixing problems — salt flowing into and out of a tank.
- Drug concentration in the bloodstream with continuous infusion.
Your turn
What is the integrating factor for dy/dx + (1/x)y = x?