(a)

Step 1:

The differential equation is .

The initial condition is and -value is .

Step size is .

Euler 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,

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.

(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 .

(c)

Step 1:

The differential equation is .

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

The exact solution is .

From the exact solution, the particular solution :

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 is almost same in both methods.

 Solution:

The particular solution at x = 2 is 3.