Question

If \[ \frac{dy}{dx}=xy+x,\; x\in\mathbb{R} \] and \(y(0)=1\), then \(y(0.2)\) with step size \(h=0.1\) using Euler's method is

• A. \(1.01\)
• B. \(1.1\)
• C. \(1.02\)
• D. \(1.2\)

Hint:

In Euler's method, always update the values sequentially using \[ y_{n+1}=y_n+h\,f(x_n,y_n). \] Never substitute the exact value of the solution while computing the next approximation.

Answer 1

The Correct Option is C

Concept: Euler's method is a numerical technique for approximating the solution of the initial value problem \[ \frac{dy}{dx}=f(x,y), \qquad y(x_0)=y_0. \] The iterative formula is \[ y_{n+1}=y_n+h\,f(x_n,y_n), \] where \(h\) is the step size.

Step 1: Write the given data. Given, \[ \frac{dy}{dx}=xy+x=x(y+1), \] with \[ x_0=0,\qquad y_0=1,\qquad h=0.1. \]

Step 2: Compute \(y_1\). First, \[ f(x_0,y_0)=0(1+1)=0. \] Hence, \[ y_1 = y_0+h\,f(x_0,y_0) = 1+0.1(0) = 1. \] Thus, \[ x_1=0.1,\qquad y_1=1. \]

Step 3: Compute \(y_2\). Now, \[ f(x_1,y_1) = 0.1(1+1) = 0.2. \] Therefore, \[ y_2 = 1+0.1(0.2) = 1.02. \] Hence, \[ \boxed{y(0.2)\approx1.02.} \] Therefore, the correct option is \[ \boxed{(C)\;1.02.} \]