Question

Consider $ \frac{dy}{dx} = x^2 - y, y(0)=1 $ using Euler's method with step size of 0.1. Then the value of $ y(0.1) $ is

• A. \(0.6\)
• B. \(0.7\)
• C. \(0.8\)
• D. \(0.9\)

Hint:

Euler's method is a fundamental numerical technique. It's crucial to correctly identify \( f(x,y) \), the initial conditions \( (x_0, y_0) \), and the step size \( h \). Remember that the formula is \( y_{n+1} = y_n + h \cdot (\text{slope at } (x_n, y_n)) \). Be mindful of signs and decimal point accuracy during calculations.

Answer 1

The Correct Option is D

Step 1: Understand Euler's Method and identify given values. Euler's method is a numerical technique used to approximate solutions to first-order ordinary differential equations (ODEs) with an initial condition. It approximates the solution curve by a sequence of line segments. The core formula for Euler's method is: \[ y_{n+1} = y_n + h f(x_n, y_n) \] Here's what each term represents: \( \frac{dy}{dx} = f(x,y) \): This is the differential equation itself, which defines the slope of the solution curve at any point \((x,y)\). In this problem, \( f(x,y) = x^2 - y \).
\( h \): This is the step size, representing the increment in the independent variable \(x\). A smaller \(h\) generally leads to a more accurate approximation but requires more steps. Here, \( h = 0.1 \).
\( (x_n, y_n) \): These are the coordinates of the current point on the approximate solution curve.
\( (x_{n+1}, y_{n+1}) \): These are the coordinates of the next approximated coordinates. From the problem statement, we are given:
Initial condition: \( y(0) = 1 \). This means at the starting point, \( x_0 = 0 \) and \( y_0 = 1 \).
We need to find the value of \( y(0.1) \). This implies we need to calculate \( y_1 \), which corresponds to \( x_1 = 0.1 \).
Step 2: Calculate the next \(x\) value.
The next \(x\) value, \(x_1\), is obtained by adding the step size \(h\) to the current \(x\) value, \(x_0\).
\[ x_1 = x_0 + h \] Substitute the values: \[ x_1 = 0 + 0.1 \] \[ x_1 = 0.1 \] This confirms that finding \( y_1 \) will give us \( y(0.1) \).
Step 3: Apply Euler's method formula to calculate \(y_1\).
Now, substitute the known values \( x_0, y_0 \), and \( h \) into the Euler's method formula to find \( y_1 \): \[ y_1 = y_0 + h f(x_0, y_0) \] First, evaluate \( f(x_0, y_0) \). Since \( f(x,y) = x^2 - y \): \[ f(x_0, y_0) = f(0, 1) = (0)^2 - 1 = 0 - 1 = -1 \] Now substitute this value back into the Euler's formula for \( y_1 \): \[ y_1 = 1 + (0.1) \times (-1) \] Perform the multiplication: \[ y_1 = 1 - 0.1 \] Perform the subtraction: \[ y_1 = 0.9 \] Therefore, the value of \( y(0.1) \) is \( 0.9 \). The final answer is \( \boxed{0.9} \).