Change of variables in a differential equation in Maple

[CAF123]

Homework Statement

Consider the differential equation zZ'' + Z' + γ² Z = 0, where Z = Z(z). Use the change of variables x = √(z/b) with b a constant to obtain the differential equation Z'' + (1/x)Z' + α²Z = 0, where Z = Z(x) and α= 2γ √b

Homework Equations

Maple commands

The Attempt at a Solution

I tried the following:

with(PDEtools):

PDE:= ...
tr:= {z = x^2 b}

dchange(tr, PDE)

however, this didn't work, possibly because maple is interpreting b as a variable , but I am not sure how to specify that it is simply a constant. I am not sure whether the above will give me what I want, since I have not specified anywhere that I want in the final DE, Z=Z(x).
Thanks.

 

[KataKoniK]

Hello,

Thank you for your post. To specify that b is a constant, you can use the assume command before defining your transformation:

assume(b, constant);

This will tell Maple to treat b as a constant rather than a variable. Then, you can proceed with your dchange command as you have already done.

However, as you mentioned, this may not give you the desired result since you have not specified that Z=Z(x) in the final DE. To do this, you can use the subs command to substitute z = x^2 b in the original DE:

DE := z*diff(Z(z),z$2)+diff(Z(z),z)+gamma^2*Z(z) = 0;
DE := subs(z = x^2*b, DE);

This will give you the desired DE with Z = Z(x). You can then proceed with your dchange command to obtain the final DE with the change of variables.

I hope this helps. Let me know if you have any further questions.