Hint
Check the forcing trial against the auxiliary roots. Use \(\displaystyle x e^{−ax}\) because the forcing exponent is an auxiliary root.
Worked solution
(a)
\(\displaystyle (3-1)P=9\implies P=\frac{9}{2}\)
Answer: \(\displaystyle \frac{9}{2}\).
Answer: \(\displaystyle \frac{9}{2}\)
(b)
\(\displaystyle A+B=7,\ -1A+P-3B=8\implies A=\frac{49}{4}\)
Answer: \(\displaystyle \frac{49}{4}\).
Answer: \(\displaystyle \frac{49}{4}\)
(c)
\(\displaystyle B=y(0)-A=- \frac{21}{4}\)
Answer: \(\displaystyle - \frac{21}{4}\).
Answer: \(\displaystyle - \frac{21}{4}\)
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: Use \(\displaystyle x e^{−ax}\) because the forcing exponent is an auxiliary root.
- 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 omit the complementary function.
(b) · 2 marks
- 1 mark: Apply both initial conditions after adding the resonant particular integral.
- 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: A resonant exponential trial needs an additional x factor.
(c) · 2 marks
- 1 mark: Use the initial value equation.
- 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 omit the complementary function.
Total: 6 marks.