PDE Separation of Variables: dT/dz+(1/x)*dT/dy+d^2T/dx^2=0

jean15paul

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I'm an engineering grad student and my professor gave me this excercise. I don't even have to solve this PDE just seperate the variables (multiplicative, not additive).

dT/dz + (1/x)*dT/dy + d[sup:251852j2]2[/sup:251852j2]T/dx[sup:251852j2]2[/sup:251852j2] = 0

where T(x,y,z) = X(x)*Y(y)*Z(z)

This is what I've done so far:

(plug in definition of T(x,y,z) above)
XYZ' + (1/x) * XY'Z + X''YZ = 0

(divide all terms by XYZ)
Z'/Z + (1/x) Y'/Y + X''/X = 0

Now I can't figure out how to get the (1/x) with only the X''/X term.

Can anyone help?
 
One of the solution would be dT/dy = 0, thus

T = X(x) * Z(z)
 
Subhotosh Khan said:
One of the solution would be dT/dy = 0, thus

T = X(x) * Z(z)

Thanks for the reply. I really appreciate the help. But I don't think this is what my professor is looking for.

Assuming T is a separable function of x, y, & z in the form T=X(x)Y(y)Z(z), I have to separate all the variables to be with like terms ... so I have to have a term with all x's, a term with all y's, and a term with all z's. I can't figure out how to get the (1/x) away from the Y'/Y and only with the X'/X term. If I multiply everything by x then I get:

x*Z'/Z + Y'/Y + x*X'/X = 0

so that doesn't work because I still have an x with the Z'/Z term. I need to get all the x's together.

Any more thoughts?
 
jean15paul said:
Subhotosh Khan said:
One of the solution would be dT/dy = 0, thus

T = X(x) * Z(z)

Thanks for the reply. I really appreciate the help. But I don't think this is what my professor is looking for.

Assuming T is a separable function of x, y, & z in the form T=X(x)Y(y)Z(z), I have to separate all the variables to be with like terms ... so I have to have a term with all x's, a term with all y's, and a term with all z's. I can't figure out how to get the (1/x) away from the Y'/Y and only with the X'/X term. If I multiply everything by x then I get:

x*Z'/Z + Y'/Y + x*X'/X = 0

so that doesn't work because I still have an x with the Z'/Z term. I need to get all the x's together.

Any more thoughts?

If you assume

dT/dy = 0

then the PDE reduces to

dT/dz + d^2T/dZ^2 = 0

Now you would assume the solution to be

T(x,z) = X(x) * Z(z) ---> No y dependancy

and finally

X"/x + Z'/z = 0
 
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