- #1
notReallyHere
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Hello.
Let g(x,y) be a function that has second order partial derivatives. Transform the differential equation
[tex]\frac{\delta ^{2}g}{\delta x^{2}}-\frac{\delta ^{2}g}{\delta y^{2}}=xyg[/tex]
by chaning to the new variables [tex]u=x^2-y^2[/tex] and [tex]v=xy[/tex]
The equation doesn't have to be solved.
Okay, so this is basically an exercise in using the chain rule.
The first part is pretty easy.
A
[tex]\frac{\delta g}{\delta x}=\frac{\delta g}{\delta u}\cdot \frac{\delta u}{\delta x}+\frac{\delta g}{\delta v}\cdot \frac{\delta v}{\delta x}=2x\cdot \frac{\delta g}{\delta u}+y\frac{\delta g}{\delta v}[/tex]
and
B
[tex]\frac{\delta g}{\delta y}=\frac{\delta g}{\delta u}\cdot \frac{\delta u}{\delta y}+\frac{\delta g}{\delta v}\cdot \frac{\delta v}{\delta y}=-2y\cdot \frac{\delta g}{\delta u}+x\frac{\delta g}{\delta v}[/tex]
Now I try to find [tex]\frac{\delta ^{2}g}{\delta x^{2}}[/tex] using the information from A. I realize I'm supposed to use the chain rule again, but I just can't get the right answer, so I guess I'm missing some fundamental understanding of what's going on here.
Here's what the first part of the problem eventually should boil down to:
[tex]\frac{\delta ^{2}g}{\delta x^{2}}=2\frac{\delta g}{\delta u}+4y^{2}\cdot \frac{\delta^{2} g}{\delta u^{2}}+4xy\cdot \frac{\delta^{2} g}{\delta u\delta v}+y^{2}\cdot \frac{\delta^{2} g}{\delta v^{2}}[/tex]
This part above is what I need help with, if I understand that part, I think I can manage to solve the rest of the problem.
Help is much appreciated. If I find out how to do it I will post here later. Thanks.
Let g(x,y) be a function that has second order partial derivatives. Transform the differential equation
[tex]\frac{\delta ^{2}g}{\delta x^{2}}-\frac{\delta ^{2}g}{\delta y^{2}}=xyg[/tex]
by chaning to the new variables [tex]u=x^2-y^2[/tex] and [tex]v=xy[/tex]
The equation doesn't have to be solved.
Okay, so this is basically an exercise in using the chain rule.
The first part is pretty easy.
A
[tex]\frac{\delta g}{\delta x}=\frac{\delta g}{\delta u}\cdot \frac{\delta u}{\delta x}+\frac{\delta g}{\delta v}\cdot \frac{\delta v}{\delta x}=2x\cdot \frac{\delta g}{\delta u}+y\frac{\delta g}{\delta v}[/tex]
and
B
[tex]\frac{\delta g}{\delta y}=\frac{\delta g}{\delta u}\cdot \frac{\delta u}{\delta y}+\frac{\delta g}{\delta v}\cdot \frac{\delta v}{\delta y}=-2y\cdot \frac{\delta g}{\delta u}+x\frac{\delta g}{\delta v}[/tex]
Now I try to find [tex]\frac{\delta ^{2}g}{\delta x^{2}}[/tex] using the information from A. I realize I'm supposed to use the chain rule again, but I just can't get the right answer, so I guess I'm missing some fundamental understanding of what's going on here.
Here's what the first part of the problem eventually should boil down to:
[tex]\frac{\delta ^{2}g}{\delta x^{2}}=2\frac{\delta g}{\delta u}+4y^{2}\cdot \frac{\delta^{2} g}{\delta u^{2}}+4xy\cdot \frac{\delta^{2} g}{\delta u\delta v}+y^{2}\cdot \frac{\delta^{2} g}{\delta v^{2}}[/tex]
This part above is what I need help with, if I understand that part, I think I can manage to solve the rest of the problem.
Help is much appreciated. If I find out how to do it I will post here later. Thanks.