OCR FP3 2013 January — Question 3 8 marks

Exam BoardOCR
ModuleFP3 (Further Pure Mathematics 3)
Year2013
SessionJanuary
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicFirst order differential equations (integrating factor)
TypeStandard linear first order - variable coefficients
DifficultyStandard +0.8 This is a standard integrating factor problem from Further Maths, requiring division by x to get standard form, finding integrating factor x^(-3), and integrating x·e^(2x) by parts. The further maths context and integration by parts requirement place it moderately above average difficulty, though the method itself is routine for FP3 students.
Spec4.10c Integrating factor: first order equations
3 Solve the differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = x ^ { 4 } \mathrm { e } ^ { 2 x }\) for \(y\) in terms of \(x\), given that \(y = 0\) when \(x = 1\).
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(\frac{dy}{dx}-3\frac{y}{x}=x^3e^{2x}\)M1 Divide by \(x\)
\(I=\exp\left(\int-\frac{3}{x}dx\right)=e^{-3\ln x}\)M1
\(=x^{-3}\)A1
\(x^{-3}\frac{dy}{dx}-3x^{-4}y=e^{2x}\)M1 Multiply and recognise derivative
\(\frac{d}{dx}(x^{-3}y)=e^{2x}\)M1 Integrate
\(x^{-3}y=\frac{1}{2}e^{2x}+A\)A1
\(x=1, y=0 \Rightarrow A=-\frac{1}{2}e^2\)M1 Use condition
\(y=\frac{1}{2}x^3(e^{2x}-e^2)\)A1 
# Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx}-3\frac{y}{x}=x^3e^{2x}$ | M1 | Divide by $x$ |
| $I=\exp\left(\int-\frac{3}{x}dx\right)=e^{-3\ln x}$ | M1 | |
| $=x^{-3}$ | A1 | |
| $x^{-3}\frac{dy}{dx}-3x^{-4}y=e^{2x}$ | M1 | Multiply and recognise derivative |
| $\frac{d}{dx}(x^{-3}y)=e^{2x}$ | M1 | Integrate |
| $x^{-3}y=\frac{1}{2}e^{2x}+A$ | A1 | |
| $x=1, y=0 \Rightarrow A=-\frac{1}{2}e^2$ | M1 | Use condition |
| $y=\frac{1}{2}x^3(e^{2x}-e^2)$ | A1 | |

---
3 Solve the differential equation $x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = x ^ { 4 } \mathrm { e } ^ { 2 x }$ for $y$ in terms of $x$, given that $y = 0$ when $x = 1$.

\hfill \mbox{\textit{OCR FP3 2013 Q3 [8]}}
This paper (8 questions)
View full paper
Q1 7 Q2 6 Q3 8 Q4 7 Q5 7 Q6 11 Q7 12 Q8 14