Transforming to a Normal Form (PDE)

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Homework Statement
Find the type, transform to normal form and solve. Show your work in detail: Uxx+2Uxy+Uyy=0
Relevant Equations
Hyperbolic- AC-B^2<0
Parabolic- AC-B^2=0
Elliptic- AC-B^2>0
I don't know how to solve for u(x,y) from where I left of after 5.

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Shouldn't you be getting: ##u_{ww}=0##, i.e. ##u(w,v)=Aw+B(v)## where ##B(v)## is a function of ##v## and ##A## is a constant?
 
MathematicalPhysicist said:
Shouldn't you be getting: ##u_{ww}=0##, i.e. ##u(w,v)=Aw+B(v)## where ##B(v)## is a function of ##v## and ##A## is a constant?

The indefinite integral of 0 with respect to w is an arbitrary function of v, not an arbitrary ocnstant.
 
pasmith said:
The indefinite integral of 0 with respect to w is an arbitrary function of v, not an arbitrary ocnstant.
yes correct then A should be A(v).
 
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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