QCE Specialist Maths: Differential equations: slope field and separation Q169

Q169 · Practice questionTechnology-activeComplex unfamiliar8 marks

QUESTION 169 (8 marks)

A function satisfies

\frac{dy}{dx}=x(1-y^2),\qquad y(0)=0.

A slope field for this equation is shown. Consider the solution on

-2\le x\le2

Slope field of y prime=x(1-y squared), x from -2 to 2 and y from -0.25 to 1.25. Segments are horizontal on x=0 and y=1. Only the initial point (0,0), not the solution curve, is marked.

Solve the initial-value problem by separation of variables.

[4 marks]

Determine and classify the stationary point of this solution.

[2 marks]

Determine all values of

x

in the interval for which

y=1/2

. Give exact values and decimals to four decimal places.

[2 marks]

Question linkSyllabus coverage

WORKED SOLUTION

Practice marking scheme

8 marks

ANSWER

(a)

y=(e^{x^2}-1)/(e^{x^2}+1)

. (b) Unique minimum at

(0,0)

. (c)

x=\pm\sqrt{\ln3}\approx\pm1.0481

Worked solution

(a) In the region

-1<y<1

, separate variables:

\int\frac{dy}{1-y^2}=\int x\,dx.

Partial fractions give

\frac12\ln\left(\frac{1+y}{1-y}\right)=\frac{x^2}{2}+C.

The initial condition gives

C=0

. Solving

\ln((1+y)/(1-y))=x^2

yields

y=\frac{e^{x^2}-1}{e^{x^2}+1}.

This solution stays in

[0,1)

for all finite real

x

, so the separation is valid on the stated interval.

(b) Here

1-y^2>0

, so the derivative has the sign of

x

. The curve decreases for

x<0

and increases for

x>0

, with a unique minimum at

(0,0)

y=1/2

(1+y)/(1-y)=3=e^{x^2}

. Therefore

x=\pm\sqrt{\ln3}\approx\pm1.0481

, both in the interval. The solution is even; this does not require its derivative to be even.

Equivalent mathematically justified methods accepted.

Exact answers unless a decimal accuracy is specified.

Part a: Separate the variables correctly.

[1 mark]

Part a: Integrate using correct partial fractions or equivalent.

[1 mark]

Part a: Apply y(0)=0 to obtain the integration constant.

[1 mark]

Part a: Solve explicitly for y and ensure the expression is valid.

[1 mark]

Part b: Find the sole stationary point (0,0).

[1 mark]

Part b: Use the derivative sign change to identify a minimum.

[1 mark]

Part c: Obtain x squared=ln(3).

[1 mark]

Part c: Give both signed roots and decimals.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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Differential equations: slope field and separation — Question 169

QUESTION 169 (8 marks)

Practice marking scheme