Pre-U Pre-U 9794/2 2017 June — Question 6 7 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2017
SessionJune
Marks7
TopicDifferential equations
TypeSeparable variables - standard (polynomial/exponential x-side)
DifficultyModerate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (using substitution for the right side), apply initial condition, and rearrange. The algebra is clean and the integration is routine for A-level, making it slightly easier than average but still requiring proper execution of the method.
Spec1.08k Separable differential equations: dy/dx = f(x)g(y)
6 Find the solution of the differential equation $$x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x + 1$$ given that \(y = 3\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).
Question 6: Differential equation \(xy^2\frac{\mathrm{d}y}{\mathrm{d}x} = x+1\)
- \(\int y^2\,\mathrm{d}y = \int\frac{x+1}{x}\,\mathrm{d}x\) M1\* Separate variables
- \(= \int 1 + \frac{1}{x}\,\mathrm{d}x\) M1 Attempt to deal with improper fraction (could include integration by parts)
- A1 Correct useable expression
- \(\frac{1}{3}y^3 = \ldots\) A1 Correct LHS
AnswerMarks Guidance
- \(x + \lnx + c\) A1 Correct RHS
- \(9 = 1 + \ln 1 + c\) M1d\* Substitute \(x=1\), \(y=3\) to find \(c\)
AnswerMarks Guidance
- \(y = \sqrt[3]{3(x + \lnx + 8)}\) A1 Obtain correct equation, in required form. Allow \(\ln x\) without modulus sign
Total: 7 marks
**Question 6: Differential equation $xy^2\frac{\mathrm{d}y}{\mathrm{d}x} = x+1$**

- $\int y^2\,\mathrm{d}y = \int\frac{x+1}{x}\,\mathrm{d}x$ **M1\*** Separate variables
- $= \int 1 + \frac{1}{x}\,\mathrm{d}x$ **M1** Attempt to deal with improper fraction (could include integration by parts)
- **A1** Correct useable expression
- $\frac{1}{3}y^3 = \ldots$ **A1** Correct LHS
- $x + \ln|x| + c$ **A1** Correct RHS
- $9 = 1 + \ln 1 + c$ **M1d\*** Substitute $x=1$, $y=3$ to find $c$
- $y = \sqrt[3]{3(x + \ln|x| + 8)}$ **A1** Obtain correct equation, in required form. Allow $\ln x$ without modulus sign

**Total: 7 marks**
6 Find the solution of the differential equation

$$x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x + 1$$

given that $y = 3$ when $x = 1$. Give your answer in the form $y = \mathrm { f } ( x )$.

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2017 Q6 [7]}}
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