Solutions Not Expressible in Terms of Elementary Functions. As discussed in calculus...

Solutions Not Expressible in Terms of Elementary Functions. As discussed in calculus, certain indefinite integrals (antiderivatives) such as cannot be expressed in finite terms using elementary functions. When such an integral is encountered while solving a differential equation, it is often helpful to use definite integration (integrals with variable upper limit). For example, consider the initial value problem

The differential equation separates if we divide by y² and multiply by dx.We integrate the separated equation from x = 2 to x = x₁ and find

If we let t be the variable of integration and replace x1 by x and y (2) by 1, then we can express the solution to the initial value problem by

Use definite integration to find an explicit solution to the initial value problems in parts (a)–(c).

(d) Use a numerical integration algorithm (such as Simpson’s rule, described in Appendix C) to approximate the solution to part (b) at x = 0.5 to three decimal places