Problem Statement
Consider the differential equation $\frac{dy}{dx} = x + y^2$ with initial condition $y(0) = 1$.
**(a)** [6 marks] Use Euler's method with a step size of $h = 0.1$ to find an approximation for $y(0.3)$, showing your values in a table to four decimal places.
Verified Solution & Marking Scheme
Part (a) [M1]
Step 1 ($x_0 = 0, y_0 = 1$): $f(0, 1) = 0 + 1^2 = 1.0000$. $y_1 = 1 + 0.1(1) = 1.1000$
Part (a) [A1]
Step 2 ($x_1 = 0.1, y_1 = 1.1$): $f(0.1, 1.1) = 0.1 + 1.1^2 = 0.1 + 1.21 = 1.3100$
Part (a) [M1]
$y_2 = 1.1 + 0.1(1.31) = 1.1 + 0.131 = 1.2310$
Part (a) [A1]
Step 3 ($x_2 = 0.2, y_2 = 1.2310$): $f(0.2, 1.231) = 0.2 + (1.231)^2 = 0.2 + 1.51536 = 1.7154$
Part (a) [A1]
$y_3 = 1.2310 + 0.1(1.7154) = 1.2310 + 0.17154 = 1.4025$
Part (a) [A1]
Approximation for $y(0.3) \approx 1.40$ (or $1.4025$)