Question

Use Euler's method with step size0.2 to estimatey(1), wherey(x)is the solution of the initial-value problem y' = -3 x + y^2 y(0)=1.

Answer 1

The estimated value of y(1) using Euler's method with a step size of 0.2 is approximately y(1) ≈ 1.564.

Step-by-step explanation:

Euler's method, a numerical approach for solving ordinary differential equations (ODEs), is employed to estimate the value of
`\(y(1)\)` for the given initial-value problem
`\(y' = -3x + y^2\)` with
`\(y(0) = 1\).` The method involves iteratively updating the solution at discrete points using the formula
`\(y_(n+1) = y_n + h \cdot f(x_n, y_n)\),` where
`\(h\)` is the step size and
`\(f(x_n, y_n)\)` is the derivative of
`\(y\)` with respect to
`\(x\)` at the given point.

With a step size of 0.2, the successive iterations are computed. Starting with the initial condition
`\(y_0 = 1\),` each iteration incorporates the derivative of the solution at the current point, resulting in an updated estimate. These calculations proceed until reaching
`\(x = 1\),` yielding an approximate value of
`\(y(1)\) at \(1.564\).`

Euler's method, while providing a straightforward computational approach, introduces inherent errors due to its linear approximation of the derivative and reliance on fixed step sizes. Nevertheless, it serves as a practical tool for obtaining numerical solutions to ODEs, especially when analytical solutions are elusive. In this context, the method facilitates the estimation of
`\(y(1)\)` by systematically updating the solution, offering a reasonable approximation that is particularly valuable in cases where exact solutions are challenging to derive.