Second order equations: particular integrals

General solution = CF + PI

If y = u is the complementary function (solving the equation with 0 on the right) and y = v is any particular integral, then y = u + v is the general solution. It has two arbitrary constants, both in the CF.

You should know the terms complementary function and particular integral.

Trial functions

Choose a trial with unknown coefficients, substitute, and compare coefficients.

Standard trial functions

When the trial is already in the CF

If the trial function is part of the CF, it gives 0 when substituted, so it cannot work. Multiply the trial by x (by x² if the CF has a repeated root containing that term).

Example: y'' - 3y' + 2y = e^2x has CF Ae^x + Be^2x, so try λxe^2x.

Exam technique

Usually write down the CF first, because the right trial PI depends on it. For a trig right-hand side, always include both cos and sin in the trial, even if only one appears in f(x). If f(x) is a sum, find a PI for each part and add them. If the question gives you the form of the PI, use it: do not find the CF unless asked (examiners' report, June 2025).

Worked example

Find the general solution of y'' - 5y' + 6y = 12x + 2.

1. CF: m² - 5m + 6 = 0, m = 2, 3, so y = Ae^2x + Be^3x.
2. Try y = λx + μ: y' = λ, y'' = 0. Substitute: -5λ + 6λx + 6μ = 12x + 2.
3. x terms: 6λ = 12, λ = 2. Constants: -10 + 6μ = 2, μ = 2.

Answer: y = Ae^2x + Be^3x + 2x + 2

Check yourself

1. Find λ so that y = λe^3x is a particular integral of y'' + y' - 2y = 4e^3x. Give your answer as a fraction in its simplest form.
   Show answer

   2/5. Substitute: 9λ + 3λ - 2λ = 10λ = 4, so λ = 2/5.

2. Which trial function should be used for a particular integral of y'' - 3y' + 2y = e^2x?
   • λxe^2x
   • λe^2x
   • λx² e^2x
   • λe^x
   Show answer

   λxe^2x. The CF is Ae^x + Be^2x. Since e^2x is in the CF, multiply by x.

3. y = λ cos x + μ sin x is a particular integral of y'' + 4y = 10cos x. Find λ. Give your answer as a fraction in its simplest form.
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   10/3. y'' = -λ cos x - μ sin x. Substitute: 3λ cos x + 3μ sin x = 10cos x, so λ = 10/3 and μ = 0.

4. Which trial function should be used for a particular integral of y'' + 4y = sin 2x?
   • λ cos 2x + μ sin 2x
   • x(λ cos 2x + μ sin 2x)
   • λ sin 2x
   • λx sin 2x
   Show answer

   x(λ cos 2x + μ sin 2x). The CF is A cos 2x + B sin 2x, which contains the right-hand side, so multiply the trial by x.

5. A particular integral of y'' + 2y' + y = x² has the form ax² + bx + c. Find c.
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   6. y' = 2ax + b, y'' = 2a. Substitute: 2a + 4ax + 2b + ax² + bx + c = x². So a = 1; 4a + b = 0 gives b = -4; 2a + 2b + c = 0 gives c = 6.

6. Which trial function should be used for a particular integral of y'' - 4y' + 4y = e^2x?
   • λxe^2x
   • λx² e^2x
   • λe^2x
   • λx³ e^2x
   Show answer

   λx² e^2x. The CF is (A + Bx)e^2x, which contains both e^2x and xe^2x. So multiply by x².

7. y = λxe^x is a particular integral of y'' - y = 3e^x. Find λ. Give your answer as a fraction in its simplest form.
   Show answer

   3/2. y' = λ(1 + x)e^x, y'' = λ(2 + x)e^x. Then y'' - y = 2λe^x = 3e^x, so λ = 3/2.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit F2
• Pearson Edexcel IAL Further Pure Mathematics 2 scheme of work
• Pearson Edexcel IAL Further Pure Mathematics F2 (WFM02) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.