Integrating $sin x^2$?

Question

On page 52 of Differential equations by Zill there is a question to find an explicit solution of the given initial-value problem.

$$\text{30. }\frac{dy}{dx} = y^2 \sin(x^2) ,\qquad y(-2) = \frac{1}{3}$$

so I seperated the $y$'s from the $x$'s and took the integral of the $y$ side but can't figure out the $\sin(x^2)$ integral

$$\frac{-1}{y} + C = \int \sin(x^2)dx $$

Anyone know how to do that?

The improper definite integral over $[0,\infty)$ can be computed, as in this answer, but there is no elementary anti-derivative. — robjohn, Sep 30 '15 at 23:18

score 1 · Answer 1 · answered Sep 30 '15 at 23:49

1

$$y = \frac{1}{3-\int_{-2}^x \sin(u^2) \, du}$$

Presumably, the point is that you should still solve for $y$ given the boundary condition, even though the integral cannot be evaluated in terms of elementary functions.

answered Sep 30 '15 at 23:49

jcandy

280

Trying it this way I end up with sin(u^2) instead of sin(x^2) which I'm still not sure how to figure out, if I use sin(u) with u=x^2, would that work? – Givanildo Souza Oct 01 '15 at 00:29
1

Sorry but I don't quite follow. To be clear, u is a dummy variable here, and the definite integral is a function of x, making y=y(x) a specific function of x. This is as simple as you can make the answer without resorting to special functions or a numerical evaluation. – jcandy Oct 01 '15 at 06:11
Look up for FrenselS integral. – miyagi_do Oct 02 '15 at 03:24

Integrating $sin x^2$?

1 Answers1