Question

Consider the following ordinary differential equation: \[ \frac{dy}{dx} = x^2y \] The initial value is \( y(0) = 1 \) and the step-size is 0.1. Solving this differential equation by Euler’s first-order method, the value of \( y(0.2) \) is ____________ (rounded off to three decimal places).

Hint:

Euler's method is an approximation technique for solving ordinary differential equations by using discrete steps.

Answer 1

Correct Answer: 1.001

To solve the differential equation using Euler's method, we use the formula: \[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \] where \( f(x, y) = x^2y \) and \( h = 0.1 \).
We are given the initial condition \( y(0) = 1 \) and want to find \( y(0.2) \). First, we calculate \( y(0.1) \): \[ y(0.1) = y(0) + 0.1 \cdot (0^2 \cdot 1) = 1 + 0 = 1 \] Next, calculate \( y(0.2) \): \[ y(0.2) = y(0.1) + 0.1 \cdot (0.1^2 \cdot 1) = 1 + 0.1 \cdot (0.01) = 1 + 0.001 = 1.001 \] Thus, the value of \( y(0.2) \) is 1.001.
Final Answer: 1.001