Physics I / Differential models in one variable

Practice question · Put in order

Order the steps to solve a linear first-order ODE y' + P(x) $y = Q(x).$

Write in standard form: dy/dx + P(x) $y = Q(x)$
Compute integrating factor $\mu$ (x) = e^( $\int P(x)dx$ )
Integrate both sides and solve for y
Multiply both sides by $\mu$ : $d/dx[\mu(x)y] = \mu(x)Q(x)$

Hints

The integrating factor must be formed before it can be applied.
Recognising the left side as one derivative is what makes integration possible.

Show the answer

Write in standard form: dy/dx + P(x) $y = Q(x)$
Compute integrating factor $\mu$ (x) = e^( $\int P(x)dx$ )
Multiply both sides by $\mu$ : $d/dx[\mu(x)y] = \mu(x)Q(x)$
Integrate both sides and solve for y

Why

The integrating factor transforms the left side into an exact derivative, making integration straightforward.

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