Use Euler’s method with dx = 0.2 to estimate y(2) if y′ = y/x and y(1) = 2. What is the exact value of y(2)?

Verified step by step guidance

1. Identify the differential equation and initial condition: \(\frac{dy}{dx} = \frac{y}{x}\) with \(y(1) = 2\).
2. Set the step size \(\Delta x = 0.2\) and note that you want to estimate \(y(2)\) starting from \(x=1\).
3. Apply Euler's method iteratively: for each step, calculate the slope \(f(x_n, y_n) = \frac{y_n}{x_n}\), then update \(y\) using \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\) and increment \(x\) by \(\Delta x\).
4. Perform the iterations from \(x=1\) to \(x=2\) in steps of \(0.2\), updating \(y\) at each step using the formula above.
5. To find the exact value of \(y(2)\), solve the differential equation analytically by separating variables: \(\frac{dy}{y} = \frac{dx}{x}\), integrate both sides, apply the initial condition to find the constant, and then evaluate the solution at \(x=2\).

Key Concepts

Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a step size (dx) to incrementally estimate the value of the function by applying the slope given by the differential equation at each step. This method is especially useful when an exact solution is difficult to find.

Solving First-Order Differential Equations

A first-order differential equation relates a function and its derivative. To solve it exactly, one often separates variables or uses an integrating factor. For the equation y' = y/x, separating variables allows integration to find the explicit formula for y(x), which can then be evaluated at any point.

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a particular point, such as y(1) = 2. This condition is essential for determining the unique solution to a differential equation. Both numerical methods like Euler's and exact solutions rely on this initial condition to find or approximate y at other points.