Further Mathematics

1. 1.

   Solve y'' - 3y' + 2y = 0 with y(0) = 3 and y'(0) = 4.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The linear constant-coefficient equation reduces to m^2 - 3m + 2 = 0.
   2. Differentiate the general solution before applying the second condition; subtraction gives B = 1.

   Marking points

   • Characteristic roots are 1 and 2.
   • y = A e^x + B e^(2x), with A + B = 3 and A + 2B = 4.
   • y = 2e^x + e^(2x).

   Examiner tip: Use y' rather than y for the velocity-like initial condition.

2. 2.

   Find the general solution of y'' - 4y' + 4y = 0 and explain why two constants are needed.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Writing A e^(2x) + B e^(2x) would only yield one effective constant.
   2. The repeated root requires the extra x factor to construct a second independent solution.

   Marking points

   • The characteristic equation is (m - 2)^2 = 0.
   • y = (A + Bx)e^(2x).
   • The independent solutions e^(2x) and xe^(2x) allow two independent initial conditions.

   Examiner tip: Repeated roots need x e^(mx), not a duplicate exponential.

3. 3.

   Solve y'' + 2y' + 5y = 0 with y(0) = 1 and y'(0) = 0. State the long-term behaviour.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The negative real part supplies a decaying envelope; the imaginary part supplies oscillation.
   2. Apply the product rule to the exponential envelope when calculating y'(0).

   Marking points

   • Roots are -1 +/- 2i, giving y = e^(-x)(A cos 2x + B sin 2x).
   • A = 1 and -A + 2B = 0, so B = 1/2.
   • y = e^(-x)(cos 2x + sin 2x/2), tending to zero as x tends to infinity.

   Examiner tip: Ignoring the derivative of e^(-x) would incorrectly give B = 0.

4. 4.

   Find the general solution of y'' - y = 6e^(2x).

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The forcing exponent two is not a characteristic root, so the usual exponential trial works.
   2. The second derivative multiplies the trial by four; subtracting the trial leaves three times it.

   Marking points

   • Complementary solution: A e^x + B e^(-x).
   • Try C e^(2x); substitution gives 3C = 6.
   • y = A e^x + B e^(-x) + 2e^(2x).

   Examiner tip: Include the complementary solution as well as the particular integral.

5. 5.

   Solve y'' + 4y = 8cos(2x), with y(0) = 0 and y'(0) = 0. Explain why a trial C cos(2x) fails.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Matching the forcing frequency to the natural frequency produces resonance, so multiply the trial by x.
   2. The derivative of 2x sin(2x) is 2sin(2x) + 4x cos(2x), zero at the origin as required.

   Marking points

   • C cos(2x) is already a complementary solution and gives zero on the left.
   • Try Cx sin(2x); y'' + 4y = 4C cos(2x), so C = 2.
   • Initial conditions eliminate both complementary constants; y = 2x sin(2x).

   Examiner tip: A resonant particular integral has a growing envelope even with zero initial displacement.

6. 6.

   Find all solutions of y'' + y = 0 on [0,pi] satisfying y(0) = 0 and y(pi) = 0. Explain whether these conditions determine a unique solution.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. These are boundary conditions at different points, not an initial value and derivative at the same point.
   2. The second equation becomes 0 = 0 after the first is applied and therefore supplies no information about B.

   Marking points

   • General solution: A cos x + B sin x.
   • y(0) = 0 forces A = 0; y(pi) = 0 holds for every B.
   • All solutions are y = B sin x for real B; uniqueness fails.

   Examiner tip: Two stated conditions need not be two independent restrictions.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.