Exam-Style Question on Differential Equations

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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 $$x\dfrac{dv}{dx}=v^2$$

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(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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