Consider the initial value problem

(a) Using definite integration, show that the integrating factor for the differential equation can be written as

and that the solution to the initial value problem is

(b) Obtain an approximation to the solution at x = 1 by using numerical integration (such as Simpson’s rule, Appendix C) in a nested loop to estimate values of µ(x) and, thereby, the value of

[Hint: First, use Simpson’s rule to approximate µ (x) at x = 0.1, 0.2,…, 1. Then use these values and apply Simpson’s rule again to approximate

(c) Use Euler’s method (Section 1.4) to approximate the solution at x = 1, with step sizes h = 0.1 and 0.05.

[A direct comparison of the merits of the two numerical schemes in parts (b) and (c) is very complicated, since it should take into account the number of functional evaluations in each algorithm as well as the inherent accuracies.]

Step 1</p>

In this problem we have to that the integrating factor for the differential equation can be written as

Step 2</p>

Method of solving linear Equations

Now, we have to calculate the integrating factor

So, multiplying the standard equation by

Last step we have to integrate the last equation with respect to y and divide by

Step 3</p>

In the Question it is given that initial value problem

The given is of the form of

Now, we have to calculate the integration factor

If we let t be the variable then the upper limit be and dx change to dt

Step 4</p>

So, multiplying the standard equation by

Last step we have to integrate the last equation from limit and divide by