Question

Consider the differential equation \[ \frac{dy}{dx}=x^{2}-y,\qquad y(0)=1. \] Using the Simple Euler's Method with a step size of \(h=0.1\), the value of \(y(0.1)\) is

• A. \(0.1\)
• B. \(0.7\)
• C. \(0.9\)
• D. \(1.0\)

Hint:

Remember the Simple Euler's Method formula: \[ \boxed{ y_{n+1}=y_n+h\,f(x_n,y_n) } \] To solve numerical problems quickly:
• Identify \(x_0,\;y_0,\) and the step size \(h\).
• Compute the slope \(f(x_0,y_0)\).
• Substitute directly into Euler's formula.
• Each application of the formula advances the solution by one step of length \(h\). This is one of the most frequently asked numerical methods in engineering mathematics examinations.

Answer 1

The Correct Option is C

Concept: Euler's Method is one of the simplest numerical techniques for approximating the solution of a first-order differential equation \[ \frac{dy}{dx}=f(x,y), \] when the exact analytical solution is difficult to obtain. If the initial condition is \[ y(x_0)=y_0, \] then the approximate value at the next point is given by \[ \boxed{ y_{n+1}=y_n+h\,f(x_n,y_n), } \] where
• \(h\) is the step size,
• \(x_{n+1}=x_n+h\),
• \(f(x_n,y_n)\) is the slope evaluated at the current point. Euler's method uses the tangent at the known point to estimate the value of the solution at the next point.

Step 1: Identify the given data.
The given differential equation is \[ \frac{dy}{dx}=x^{2}-y. \] Comparing with \[ \frac{dy}{dx}=f(x,y), \] we obtain \[ f(x,y)=x^{2}-y. \] Also, \[ x_0=0,\qquad y_0=1,\qquad h=0.1. \] We are required to find the approximate value of \[ y(0.1). \]

Step 2: Calculate the slope at the initial point.
Evaluate the function at \[ (x_0,y_0)=(0,1). \] Therefore, \[ f(0,1)=0^{2}-1=-1. \] Thus, the slope of the solution curve at the initial point is \[ \boxed{-1.} \]

Step 3: Apply Euler's formula.
Euler's approximation formula is \[ y_{1}=y_{0}+h\,f(x_{0},y_{0}). \] Substituting the known values, \[ y_{1} = 1+0.1(-1). \] Hence, \[ y_{1} = 1-0.1 = 0.9. \] Since \[ x_{1}=x_{0}+h =0+0.1 =0.1, \] we obtain \[ \boxed{ y(0.1)\approx0.9. } \]

Step 4: Write the final answer.
Therefore, using the Simple Euler's Method, \[ \boxed{ y(0.1)=0.9. } \] Hence, the correct option is \[ \boxed{\text{(C) }0.9.} \]