Question

Using Euler’s modified method, find the solution of the equation :$\frac{dy}{dx} =x+\sqrt{y}$ with initial condition y = 1 at x = 0 for the name of$0 \leq x \leq 0.3$ in steps of 0.1.

Answer 1

In mathematics and computational science, the Euler method (also called forward

Euler method) is a first-order numerical procedurefor solving ordinary differential

equations (ODEs) with a given initial value.

Consider a differential equation dy/dx = f(x, y) with initialcondition y(x0)=y0

then succesive approximation of this equation can be given by:

y(n+1) = y(n) + h * f(x(n), y(n))

where h = (x(n) – x(0)) / n

h indicates step size. Choosing smaller

values of h leads to more accurate results

and more computation time.

Consider below differential equation

dy/dx = (x + y + xy)

with initial condition y(0) = 1

and step size h = 0.025.

Find y(0.1).

Solution:

f(x, y) = (x + y + xy)

x0 = 0, y0 = 1, h = 0.025

Now we can calculate y1 using Euler formula

y1 = y0 + h * f(x0, y0)

y1 = 1 + 0.025 *(0 + 1 + 0 * 1)

y1 = 1.025

y(0.025) = 1.025.

Similarly we can calculate y(0.050), y(0.075), ....y(0.1).

y(0.1) = 1.11167