Exam-Style Questions on Differential Equations

1.

A-Level

(a) Using a suitable substitution, or otherwise, find

$$ \int \frac{x}{(3x^2 - 5)^2} dx$$

(b) Solve the differential equation below giving your answer in the form \(y = f(x)\). It is given that given that y = \( \frac{1}{2} \) when x = 0.

$$ \frac{dy}{dx} = \frac{2xy^3}{(3x^2 - 5)^2}$$

Worked Solution

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2.	IB Analysis and Approaches

Consider the differential equation \( \dfrac{dy}{dx} = \dfrac{xy+y^2}{x^2} \), where \( x \gt 0, y \gt 0 \).

It is given that \(y = 3 \) when \(x = 1\).

(a) Use Euler's method with step length \(0.1\) to find an approximate value of \(y\) when \(x=1.2\).

(b) By solving the differential equation, show that \( y = \frac{-x}{ \ln |x| + C } \)

(c) Find the value of \(y\) when \(x=1.2\).

(d) With reference to the concavity of the graph of \( y = \frac{-x}{ \ln |x| + C } \) for \( 1 \le x \le 1.2\) explain why the value of \(y\) found in part (c) is greater than the approximate value of \(y\) found in part (a).

Worked Solution

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3.

A-Level

(a) Express the following fraction in partial fractions.

$$ \frac{1}{F(5-3F)} $$

The popularity of a student rock group is measured during their first year of gigs. The number of fans is modelled by the differential equation:

$$ \frac{dF}{dt} = \frac{F}{15} (5-3F) \quad 0 \le t \le 12 $$

where F, in hundreds, is the number of fans and t is the time measured in months since the band began performing regularly.

(b) Given that there were 100 fans when the measurements began, determine the time taken, in months, for the number of fans to increase by 50%.

(c) Show that:

$$ F= \frac{A}{B+C^{-\frac{t}{3}}} $$

where A, B and C are integers to be found.

Worked Solution

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4.	IB Analysis and Approaches

Consider the differential equation \(x^2\dfrac{dy}{dx}=xy+y^2\). It is given that \(y = 2\), when \(x = 1\).

(a) Use Euler's method, with a step length of 0.1, to find an approximate value of \(y\) when \(x = 1.5\).

(b) Use the substitution \(y = vx\) to show that \(x\dfrac{dv}{dx}=v^2\)

.

(c) By solving the differential equation, show that \(y = \dfrac{2x}{1-\ln{x^2}}\).

(d) Find the actual value of \(y\) when \(x = 1.5\).

(e) Using the graph of \(y = \dfrac{2x}{1-\ln{x^2}}\), suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of \(y\) at x = \(1.5\).

Worked Solution

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