Hint
Put the equation into standard linear form first. Substitute the proposed exponential particular solution.
Worked solution
(a)
\(\displaystyle (3+1)K=10\implies K=\frac{5}{2}\)
Answer: \(\displaystyle \frac{5}{2}\).
Answer: \(\displaystyle \frac{5}{2}\)
(b)
\(\displaystyle C=11-K=\frac{17}{2}\)
Answer: \(\displaystyle \frac{17}{2}\).
Answer: \(\displaystyle \frac{17}{2}\)
(c)
\(\displaystyle y=\frac{5 e^{x}}{2} + \frac{17 e^{- 3 x}}{2}\); evaluate at \(\displaystyle x=1\).
Answer: \(7.21889465227446\).
Answer: \(\approx 7.21889465227446\)
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: Substitute the proposed exponential particular solution.
- 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: The integrating factor multiplies every term.
(b) · 2 marks
- 1 mark: Apply the initial condition at \(\displaystyle x=0\).
- 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 homogeneous solution after finding a particular solution.
(c) · 2 marks
- 1 mark: Use the complete particular solution with its determined constant.
- 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: The integrating factor multiplies every term.
Total: 6 marks.