OCR FP3 Specimen — Question 1 5 marks

Exam BoardOCR
ModuleFP3 (Further Pure Mathematics 3)
SessionSpecimen
Marks5
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TopicFirst order differential equations (integrating factor)
TypeStandard linear first order - variable coefficients
DifficultyModerate -0.3 This is a straightforward application of the integrating factor method for a linear first-order ODE with standard form. The integrating factor is e^(-ln x) = 1/x, leading to routine integration. While it's a Further Maths topic, it requires only direct application of a standard technique with no problem-solving insight, making it slightly easier than average.
Spec4.10c Integrating factor: first order equations
1 Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = x ,$$ giving \(y\) in terms of \(x\) in your answer.
AnswerMarks Guidance
Integrating factor is \(e^{\int -\frac{1}{x} dx} = e^{-\ln x} = \frac{1}{x}\)M1 For finding integrating factor
A1For correct simplified form
\(\frac{d}{dx}\left(\frac{y}{x}\right) = 1 - \frac{y}{x}\) \(\int dx \Rightarrow y = x^2 + cx\)M1 For using integrating factor correctly
B1 A1For arbitrary constant introduced correctly; For correct answer in required form
5 
Integrating factor is $e^{\int -\frac{1}{x} dx} = e^{-\ln x} = \frac{1}{x}$ | M1 | For finding integrating factor
| A1 | For correct simplified form
$\frac{d}{dx}\left(\frac{y}{x}\right) = 1 - \frac{y}{x}$ $\int dx \Rightarrow y = x^2 + cx$ | M1 | For using integrating factor correctly
| B1 A1 | For arbitrary constant introduced correctly; For correct answer in required form
| **5** |
1 Find the general solution of the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = x ,$$

giving $y$ in terms of $x$ in your answer.

\hfill \mbox{\textit{OCR FP3  Q1 [5]}}
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