(a)

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, proceed in the direction indicated by the  slope.

Use a small step , move along the tangent line.

and .

Use step size , , and .

Here .

Consider .

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Substitute and .

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Construct a table for values.

The particular solution at is .

(b)

The differential equation is .

The initial condition is .

Solution to the differential equation :

Integrate on each side.

.

Substitute initial conditions , .

The exact solution is .

(c)

The differential equation is .

From Euler method the particular solution at is .

The exact solution is .

From the exact solution, find the particular solution :

Substitute in exact solution.

So the particular solution at is .

Error between Euler solution and particular is .

Therefore, the particular solution in both methods has an error of .

(a) The particular solution at is .

(b) The exact solution is .

(c) The particular solution in both methods has an error of .