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Exam-Style Question on Differential EquationsA mathematics exam-style question with a worked solution that can be revealed gradually |
Question id: 639. This question is similar to one that appeared on an IB AA Higher paper in 2022. The use of a calculator is allowed.
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 = \(1.5\).
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