Q2.6 - 17EExpert-verified
Use the method discussed under “Equations of the Form " to solve problems 17-20.
Equation of the form of for the given equation is and .
Equations of the form When the right-hand side of the equation can be expressed as a function of the combination , where a and b are constants, that is, then the substitution transforms the equation into a separable one.
Let us take
Differentiate with respect to.
Now, integrate on both sides.
Therefore, Equation of the form of for the given equation is and .
Question: 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 and, thereby, the value of .
[Hint: First, use Simpson’s rule to approximate 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.]
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