Is the Solution Unique Using the Method of Characteristics?

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Homework Statement

: Using the method of characteristics, say if the problem does have a solution; if it does, is the solution unique? [/B]
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Homework Equations


dt=(u^2)dx;

The Attempt at a Solution


from the formula above follows dt/dx=u^2; then how am I supposed to find and use characteristics?
 
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Well what is ux/ut in general? (Never sure if this is same thing as 'method of characteristics' but you can solve by asking - or by now answering - that question.)
 
No
It may be more familiar as (∂u/∂x)t/(∂u/∂t)x = ?

(I did not get that article last time I tried to read it, nor others I have tried, but will try again in next days).
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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