Approximating y(0.1) with Euler's Method

This question asks us to find an approximate value for $y(0.1)$ given the differential equation $\frac{dy}{dx} = x^2y - 1$ and the initial condition $y(0) = 1$. This is a common task in numerical analysis, where we use techniques like Euler's method to estimate solutions to differential equations when an exact analytical solution is difficult or impossible to find.

Euler's Method Explanation

Euler's method is a simple first-order numerical procedure for approximating solutions to differential equations with a given initial value. It works by taking small steps along the tangent line to the solution curve. The core formula used in Euler's method is:

$$y_{n+1} = y_n + h \cdot f(x_n, y_n)$$

Here's what each term represents:

• $(x_n, y_n)$ is the current point on the approximate solution curve.
• $h$ is the step size, which determines the size of each step taken along the x-axis ($h = x_{n+1} - x_n$). A smaller step size generally leads to a more accurate approximation but requires more calculations.
• $f(x_n, y_n)$ is the value of the derivative $\frac{dy}{dx}$ at the current point $(x_n, y_n)$, representing the slope of the solution curve at that point.
• $y_{n+1}$ is the approximate value of $y$ at the next point $x_{n+1}$.

Solving dy/dx = x^2y - 1 with Euler's

We are given:

• The function defining the slope: $f(x, y) = x^2y - 1$.
• The initial condition: $(x_0, y_0) = (0, 1)$.
• The target value for x: $x = 0.1$.

To find $y(0.1)$, we can take a single step from $x_0=0$ to $x_1=0.1$. This means our step size will be $h = x_1 - x_0 = 0.1 - 0 = 0.1$.

Initial Point Calculation for y(0.1)

First, we evaluate the slope $f(x, y)$ at the initial point $(x_0, y_0) = (0, 1)$:

$$f(x_0, y_0) = f(0, 1) = (0)^2 \cdot (1) - 1$$

$$f(0, 1) = 0 \cdot 1 - 1$$

$$f(0, 1) = -1$$

This value represents the slope of the solution curve at the starting point.

Euler's Step Calculation for y(0.1)

Now, we use the Euler's method formula to calculate the approximate value of $y$ at $x_1 = 0.1$:

$$y_1 = y_0 + h \cdot f(x_0, y_0)$$

Substitute the known values:

$$y_1 = 1 + (0.1) \cdot (-1)$$

$$y_1 = 1 - 0.1$$

$$y_1 = 0.9$$

Final Result for y(0.1) Approximation

Therefore, using Euler's method with a step size $h=0.1$, the approximate value of $y(0.1)$ for the given differential equation and initial condition is $0.900$.