First-order linear equations

Objective. Use integrating factors with initial conditions.

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Learn: the key idea

A first-order linear equation has the form dy/dx + P(x)y = Q(x), solved using an integrating factor μ(x) = e^∫P(x)dx that turns the left side into d/dx(μy). Multiply through by μ, integrate both sides, then divide by μ to isolate y, applying the initial condition to fix the constant. For dy/dx + y = x with μ = e^x, the equation becomes d/dx(e^x y) = xe^x, integrate by parts to get e^x y = xe^x − e^x + C. This method works even when the equation is not separable, provided it stays linear in y.

Worked example

Solve the first-order linear equation dy/dx + 2y = 4 with y(0) = 5.

1 $Integrating factor: μ = e^(2x)$
2 $Form d/dx(e^(2x)y) = 4e^(2x)$
3 $Integrate: e^(2x)y = 2e^(2x) + C$
4 $Use y(0)=5: C=3, so y = 2 + 3e^(-2x)$

Answer: y = 2 + 3e^(−2x)

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Common mistake. Forgetting to multiply the right-hand side Q(x) by the integrating factor as well as the left side; both sides must be multiplied by μ(x) before integrating, or the solution is wrong.

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