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MathidiaMathematics Academy
Differential equations
Lesson 37 of 75
Easy practice

First-order linear equations

Objective. Use integrating factors systematically.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A first-order linear differential equation of the form dy/dx + P(x)y = Q(x) is solved using an integrating factor I(x) = e^(∫P(x)dx), which makes the left-hand side become the derivative of I(x)y when the equation is multiplied through by I(x). Once in this form, both sides are integrated directly, and the equation solved for y. For example, for dy/dx + 2y = e^x, the integrating factor is e^(2x), turning the equation into d/dx(e^(2x)y) = e^(3x), which integrates directly.

Worked example

Use an integrating factor to solve the linear differential equation dy/dx + (1/x)y = x, for x > 0.

1 … 0
Coordinate grid with the line 1 … 0
  1. 1Here P(x)=1/x, so the integrating factor is I(x) = e^(∫(1/x)dx) = e^(ln x) = x
  2. 2Multiply through by x: x(dy/dx) + y = x², which is d/dx(xy) = x²
  3. 3Integrate both sides: xy = x³/3 + c
  4. 4Solve for y: y = x²/3 + c/x

Answer: y = x²/3 + c/x

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Common mistake. Students forget to multiply every term of the original equation (including Q(x)) by the integrating factor, only applying it to the left-hand side. The integrating factor must multiply both sides before integrating.

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