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MathidiaMathematics Academy
Differential equations
Lesson 39 of 75
Challenge

Particular integrals

Objective. Construct complementary and particular solutions.

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

Learn: the key idea

To solve a non-homogeneous second-order differential equation ay''+by'+cy=f(x), the general solution is the sum of the complementary function (the general solution of the homogeneous equation) and a particular integral (any single solution matching the right-hand side f(x)). The particular integral's trial form matches the type of f(x): a polynomial trial for polynomial f(x), an exponential trial for exponential f(x), and sine/cosine trials for trigonometric f(x), adjusted if the trial duplicates the complementary function. For example, for y''−y=4, trying y=k gives −k=4 so k=−4, and the particular integral is y=−4.

Worked example

Find the particular integral for the differential equation y'' − 3y' + 2y = 10e^(3x).

3 … 2
Coordinate grid with the line 3 … 2
  1. 1Since e^(3x) is not a solution of the homogeneous equation (roots are m=1,2), try y = ke^(3x)
  2. 2Then y' = 3ke^(3x), y'' = 9ke^(3x)
  3. 3Substitute: 9ke^(3x) − 9ke^(3x) + 2ke^(3x) = 10e^(3x), giving 2ke^(3x) = 10e^(3x)
  4. 4Solve: k = 5, so the particular integral is y = 5e^(3x)

Answer: y = 5e^(3x)

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Common mistake. Students add the complementary function and particular integral's constants together as one unknown, forgetting the complementary function already has its own arbitrary constants A and B determined separately. Keep the particular integral constant-free once found; only A and B remain arbitrary.

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