OCR MEI C4 — Question 3 4 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDifferential equations
TypeSeparable variables - standard (polynomial/exponential x-side)
DifficultyModerate -0.5 This is a straightforward separable variables question requiring standard technique: separate to get dy/y = 3x²dx, integrate both sides to ln|y| = x³ + c, then apply the initial condition. It's slightly easier than average because the separation is immediate, the integrals are basic, and there are no complications with absolute values or algebraic manipulation.
Spec1.08k Separable differential equations: dy/dx = f(x)g(y)
3 A curve satisfies the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } y\), and passes through the point \(( 1,1 )\). Find \(y\) in terms of \(x\).
Question 3:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\frac{dy}{dx} = 3x^2y \Rightarrow \int\frac{dy}{y} = \int 3x^2\,dx\)M1 separating variables
\(\Rightarrow \ln y = x^3 + c\)A1 condone absence of \(c\)
When \(x=1\), \(y=1 \Rightarrow \ln 1 = 1 + c \Rightarrow c = -1\)B1 \(c = -1\) oe
\(\Rightarrow \ln y = x^3 - 1\)
\(\Rightarrow y = e^{x^3-1}\)A1 [4] o.e. 
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 3x^2y \Rightarrow \int\frac{dy}{y} = \int 3x^2\,dx$ | M1 | separating variables |
| $\Rightarrow \ln y = x^3 + c$ | A1 | condone absence of $c$ |
| When $x=1$, $y=1 \Rightarrow \ln 1 = 1 + c \Rightarrow c = -1$ | B1 | $c = -1$ oe |
| $\Rightarrow \ln y = x^3 - 1$ | | |
| $\Rightarrow y = e^{x^3-1}$ | A1 [4] | o.e. |

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3 A curve satisfies the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } y$, and passes through the point $( 1,1 )$. Find $y$ in terms of $x$.

\hfill \mbox{\textit{OCR MEI C4  Q3 [4]}}
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Q1 8 Q2 18 Q3 4 Q4 18 Q5 20 Q6 8