Approximating Differential Equations Using Euler’s Method

00:00 4 marks

Consider the differential equation \( \frac{dy}{dt} = y – t^2 + 1 \) with the initial condition \(y(0)=0.5\).

(a) Derive the recursion formula used in Euler’s method for approximating solutions to first order differential equations.

(b) Using your derived formula with a step size of \(h=0.2\), approximate the value of \(y(0.2)\).

Clearly state each step in your reasoning and justify why Euler’s method provides a valid approximation over a small interval.

Hint Hint

Hint

Euler’s method approximates the derivative \(y'(t) \approx \frac{y(t+h)-y(t)}{h}\). Use this approximation to derive the recursion formula. For justification, consider the tangent line approximation.

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Complete Answer

Using Euler’s method, \(y(0.2)\) is approximately \(0.8\). The recursion used is \(y_{i+1}=y_i+h\,(y_i-t_i^2+1)\).

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