Exam-Style Questions.

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

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