help to finish off the general solution of a linear (quasi-linear) partial differential equation of the 2nd order with constant coefficients of the following form: Uxx+2Uxy-Uyy+Ux+Uy=0
$$U_{xx}+2*U_{xy}-U_{yy}+U_{x}+U_{y}=0$$
1^2-1(-1)=2>0 => hyperbolic type.
further, during the decision, a replacement was made
$$\xi =y+(-(1+\sqrt2))*x$$ $$\eta =y+(-(1-\sqrt2))*x$$
after being reduced to the canonical form , the equation acquired the following form:
$$U_{\xi \eta }+(\sqrt2/8)*U_{ \xi}-(\sqrt2/8)*U_{ \eta}=0$$
Also, in order to get rid of the lower derivatives, I brought this equation to a "special canonical form" (by replacement with expanent), it took the form
$$(w( \xi, \eta))_{ \xi \eta}=(-1/32)*w( \xi, \eta)$$
how to solve it. ?
I have seen an example of solving such a canonical equation: $$U_{\xi \eta }+U_{ \eta}=0$$ but I don't know how to solve my own.
there are the following questions along the way:
I am trying to make a replacement $$(w( \xi, \eta))_{\eta}=v( \xi, \eta)$$ to lower the order of the equation, but I don't know how to express it $$w ( \xi, \eta)$$ from the right side of the equation via $$v ( \xi, \eta)$$
how is the indefinite integral of the partial derivative taken? that is, I get it $$\int ( \delta w/ \delta\eta )d\eta$$ to express $$w ( \xi, \eta)$$ via $$v ( \xi, \eta)$$
I had a derivative even further out in the answer, which I could not get rid of, but I was solving it wrong there, like.
- following this topic on this forum, I tried to solve by swapping variables $$U(\xi ,\eta )=q(\xi )*p( \eta )$$ I get two primordial ones in response (the first one is from xi, the second one is from eta). How can I get rid of them? maybe I'm missing some opportunity of this in the transformation? Maybe this expression can be reduced somehow to ordinary differential equations?
In general, is it possible to solve this equation by the method I am trying to solve it - by the method of characteristics?
I have already looked at curvilinear integrals, and found the Gursa problem, where there is a similar equation, but there is another one, there are boundary/starting (I don't understand) conditions, but I still don't know.
I will be very grateful if you help
differential equations, partial differential equations, partial differential equations of the 2nd order, partial differential equations of the 2nd order with constant coefficients, partial differential equations of the 2nd order of hyperbolic type with constant coefficients, partial derivative, emf, mathphys, equations of mathematical physics