Consider the equation of the ellipse
If we introduce new variables
the ellipse becomes the circle
The change of variables (1) stretches the ellipse to a circle (or vice-verse). This change of variable acts the same way transforming
Ok, my question deals with the following PDE:
Dxx(u) + 3*Dyy(u) - 2*Dx(u) + 24*Dy(u) +5*u = 0
and there is a substitution I am instructed to use:
u = v*e^(a*x+By), and y' = d*y. Where a, B, and d are constants
I am trying to reduce the PDE down to the form Dmm(v) + Dnn(v) + c*v = 0
I don't understand what to do with the d I. have done the substitution, and and off by a factor of 3 on the Dnn(v), so I have determined that with 2 partials that d must be 1/sqrt(3), however I don't understand how it works in the problem. I know it fixes my factor of 3 problem, but I don't understand how it works.
The form at the moment is of:
Dmm(v)*e^(ax+By) +3*Dnn(v)*e^ax+By) + (a-1)*Dm(v)*e^(ax+By) + (6B+24)*Dn(v)*e^(ax+By) +(a^2 + B^2 + 24B - 2a +5)*v*e^(ax+By)
So I have determined also that a = 1 and B = -4.... But how does d come in in order to get rid of the factor of 3?
BTW sorry about not using easier notation, wasn't familiar with how it works on the forum.