find the exact solution of the differential equation analytically

Question

(a) Use Euler's Method with a step size of h = 0 .1 to approximate the particular solution of the initial value problem at the given x - value, (b) find  the exact solution of the differential equation analytically, and (c) compare the solutions at the given x - value.

Differential Equation               Initial condition                      x - value

Answer 1

(a)

Step 1:

The differential equation is .

The initial condition is .

Step size is .

Euler's method is a numerical approach to approximate the  particular solution of the differential equation.

Let that passes through the point .

From this starting point, one can proceed in the direction indicated by the  slope.

Use a small step , move along the tangent line.

and .

Step 2:

Use step size , , and .

So we have , , , ,.....and,

Answer 2

Contd....

Step 3:

Proceeding with similar calculations, we get the values in the table:

 From the table particular solution at x = 2 is 3.031.

 Solution:

The particular solution at x = 2 is 3.031.

Answer 3

(b)

Step 1:

The differential equation is .

The initial condition is .

Solution to the differential equation :

Integrate on each side.

Substitute initial conditions , .

The exact solution is .

 Solution:

The exact solution is .

Answer 4

(c)

Step 1:

The differential equation is .

From Euler's method, particular solution at x = 2 is 3.031.

From the exact solution :

The exact solution is .

Substitute x = 2 in exact solution.

Solutions to the 3^rd degree equation are .

Imaginary values are neglected.

So the particular solution at x = 2 is 3.

Therefore the particular solution at x = 2 is almost same in both the methods.

 Solution:

The particular solution at x = 2 is almost same in both the methods.