Hint
The constant-coefficient auxiliary equation applies to the stated linear equation. Solve the two linear equations from the initial conditions.
Worked solution
(a)
\(A+B=10,\ -3A-4B=11\implies A=51\)
Answer: \(51\).
Answer: \(51\)
(b)
\(B=y(0)-A=-41\)
Answer: \(-41\).
Answer: \(-41\)
(c)
\(y=51 e^{- 3 x} - 41 e^{- 4 x}\); put \(\displaystyle x=1\).
Answer: \(1.78819929232296\).
Answer: \(\approx 1.78819929232296\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Solve the two linear equations from the initial conditions.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not lose one initial condition.
(b) · 2 marks
- 1 mark: Use the initial displacement after finding A.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Differentiate the full exponential-trigonometric product when roots are complex.
(c) · 2 marks
- 1 mark: Evaluate both exponential components.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not lose one initial condition.
Total: 6 marks.