Answer to Question #158752 in Quantitative Methods for manish

Question

Use modified Euler’s method with one step to find the value of y at x = 0.1 to five significant figures, where dy/dx = x^2+y, y=0.94, when x = 0.

Answer 1

Solution

Modified Euler is a  method for numerical integration of ODE. If

y’(x) = f(x,y(x)), y(x₀)=y₀ 

the size of every step h and x_n = x₀+n*h an approximation of the solution to the ODE is

y_n+1 = y_n + h*f(x_n+h/2,y_n+h*f(x_n,y_n)/2) 

For given ODE

f(x,y) = x²+y;  y(0) = 0.94;  x₀ = 0;  y₀ = 0.94;  h = 0.1

y_n+1 = y_n + h*[ y_n +h*( y_n+ x_n²)/2+ (x_n+h/2)²)]

From this expression we’ll get  x₁ = 0.1;  y₁ = 1.03895  

Rounded to five significant figures y₁ = 1.0390

Answer

y₁ = 1.0390