1. The problem statement, all variables and given/known data
Verify if ##t=\lambda x## then ##x^2\frac{\partial^2 y}{\partial x^2} = t^2\frac{\partial^2 y}{\partial t^2}##
3. The attempt at a solution
[tex]t=\lambda x\;\Rightarrow\; \frac{\partial t}{\partial x}=\lambda[/tex]
[tex]\frac{\partial y}{\partial x} = \frac{\partial y}{\partial t}\frac{\partial t}{\partial x}= \lambda\;\frac{\partial y}{\partial t}[/tex]
[tex]\frac{\partial^2 y}{\partial x^2}=\lambda \frac{\partial^2 y}{\partial t^2} \frac{\partial t}{\partial x}=\lambda^2\frac{\partial^2 y}{\partial t^2}[/tex]
[tex]x^2\frac{\partial^2 y}{\partial x^2}=\frac{t^2}{\lambda^2}\lambda^2\frac{\partial^2 y}{\partial t^2}\;\Rightarrow\;x^2\frac{\partial^2 y}{\partial x^2} = t^2\frac{\partial^2 y}{\partial t^2}[/tex]
Am I correct?
Thanks
Verify if ##t=\lambda x## then ##x^2\frac{\partial^2 y}{\partial x^2} = t^2\frac{\partial^2 y}{\partial t^2}##
3. The attempt at a solution
[tex]t=\lambda x\;\Rightarrow\; \frac{\partial t}{\partial x}=\lambda[/tex]
[tex]\frac{\partial y}{\partial x} = \frac{\partial y}{\partial t}\frac{\partial t}{\partial x}= \lambda\;\frac{\partial y}{\partial t}[/tex]
[tex]\frac{\partial^2 y}{\partial x^2}=\lambda \frac{\partial^2 y}{\partial t^2} \frac{\partial t}{\partial x}=\lambda^2\frac{\partial^2 y}{\partial t^2}[/tex]
[tex]x^2\frac{\partial^2 y}{\partial x^2}=\frac{t^2}{\lambda^2}\lambda^2\frac{\partial^2 y}{\partial t^2}\;\Rightarrow\;x^2\frac{\partial^2 y}{\partial x^2} = t^2\frac{\partial^2 y}{\partial t^2}[/tex]
Am I correct?
Thanks
via Physics Forums RSS Feed http://www.physicsforums.com/showthread.php?t=713672&goto=newpost
0 commentaires:
Enregistrer un commentaire