Chapter 2.2, Problem 27E

Solutions Not Expressible in Terms of Elementary Functions. As discussed in calculus, certain indefinite integrals (antiderivatives) such as ${\displaystyle \int e^{x^2}dx}$ 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

$\frac{dy}{dx}=e^{x^2}{y}^2$, $y\left(2\right)=1$.

The differential equation separates if we divide by $y^2$ and multiply by $dx$. We integrate the separated equation from $x=2$ to $x=x_1$ and find

$\begin{array}{rllllllllll}{\displaystyle {\int }_x^{}}& =& 2x& =& x_1{e}^{x^2}dx& =& {\displaystyle {\int }_x^{}}& =& 2x& =& x_1\frac{dy}{y^2}\\ & =& {-\frac{1}{y}|}_x^{}& =& 2x& =& x_1\\ & =& -\frac{1}{y\left(x_1\right)}+\frac{1}{y\left(2\right)}\end{array}$

If we let $t$ be the variable of integration and replace $x_1$ by $x$ and $y\left(2\right)$ by $1$, then we can express the solution to the initial value problem by

$y\left(x\right)={\left(1-{\displaystyle {\int }_2^x{e}^{t^2}dt}\right)}^{-1}$

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

a. $dy/dx=e^{x^2}$, $y\left(0\right)=0$

b. $dy/dx=e^{x^2}{y}^{-2}$, $y\left(0\right)=1$

c. $dy/dx=\sqrt{1+\sin x}\left(1+y^2\right)$, $y\left(0\right)=1$

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.