All subjects · Further Mathematics · WFM02 Further Pure Mathematics 2

Second order equations: conditions and complete solutions

Edexcel IAL Further Mathematics · WFM02 Further Pure Mathematics 2

A full F2 question puts everything together: complementary function, particular integral, then two conditions to find the two constants. It often ends by describing how the solution behaves.

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Putting it together

1. Solve the auxiliary equation; write the CF.

2. Find the PI.

3. General solution y = CF + PI.

4. Differentiate the whole general solution (including the PI).

5. Substitute the two conditions to get two equations for A and B.

Finish with "y = ...": the June 2025 examiners' report notes a few students leaving it out.

Initial and boundary conditions

Initial conditions give y and dy/dx at the same point, often x = 0. Boundary conditions give y at two different points, for example y(0) and y(π/2).

At x = 0, e0 = 1, cos 0 = 1 and sin 0 = 0, which makes the equations simple.

Long-term behaviour

When the CF contains e-kx factors (k > 0), it dies away for large x and the solution approaches the PI. For example y = e-x(A cos x + B sin x) + 3sin x behaves like 3sin x for large x: an oscillation of amplitude 3.

Worked example

Solve y'' + 4y = 8x, given y = 1 and dy/dx = 0 when x = 0.

  1. CF: m2 + 4 = 0, m = ±2i, so y = A cos 2x + B sin 2x.
  2. PI: try λx + μ; then 4λx + 4μ = 8x, so λ = 2, μ = 0.
  3. y = A cos 2x + B sin 2x + 2x. x = 0: A = 1.
  4. dy/dx = -2A sin 2x + 2B cos 2x + 2. x = 0: 2B + 2 = 0, B = -1.

Answer: y = cos 2x - sin 2x + 2x

Key idea: CF + PI, then differentiate the whole thing and use both conditions. Always end with y = ....

Check yourself

  1. Using y = cos 2x - sin 2x + 2x from the worked example, find y when x = π/4, to 3 decimal places.
    Show answer

    0.571. cos(π/2) - sin(π/2) + π/2 = 0 - 1 + π/2 = 0.571.

  2. y'' - 3y' + 2y = 4 has general solution y = Aex + Be2x + 2. Given y = 0 and dy/dx = 1 when x = 0, find B.
    Show answer

    3. y(0): A + B + 2 = 0. y'(0): A + 2B = 1. Subtracting: B = 3, then A = -5.

  3. For the same equation, with A = -5 and B = 3, find y when x = 1, to 3 decimal places.
    Show answer

    10.576. y(1) = -5e + 3e2 + 2 = 10.576.

  4. y'' + 2y' + 5y = 0, with y = 0 and dy/dx = 4 when x = 0, has solution y = 2e-x sin 2x. Find y when x = π/4, to 4 decimal places.
    Show answer

    0.9119. y(π/4) = 2e-π/4 sin(π/2) = 2e-π/4 = 0.9119.

  5. A question gives the form of a particular integral, y = λx2 e2x, and asks you to find λ. What is the efficient method?
    • Differentiate the given form twice, substitute it into the equation, and compare coefficients
    • Find the complementary function first, then guess λ
    • Use the initial conditions to find λ
    • Integrate the right-hand side twice
    Show answer

    Differentiate the given form twice, substitute it into the equation, and compare coefficients. The June 2025 examiners' report notes some students finding the CF when the question had already given the PI's form and did not need it.

  6. Describe the behaviour of y = e-x(A cos x + B sin x) + 3sin x for large x.
    • y tends to 0
    • y grows without limit
    • y oscillates like 3sin x, with amplitude 3
    • y tends to 3
    Show answer

    y oscillates like 3sin x, with amplitude 3. The CF has the factor e-x, which tends to 0, so only the PI 3sin x remains.

  7. y'' + y = 0, with y = 2 when x = 0 and y = 5 when x = π/2. Find y when x = π/4, to 3 decimal places.
    Show answer

    4.950. y = A cos x + B sin x. y(0) = A = 2; y(π/2) = B = 5. y(π/4) = (2 + 5)/√2 = 4.950.

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