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z=A exp(pt) sin (px)

Form pde for the above equation where A and p are arbitrary constants

x ,t independent variable Z = f(x,t)

K S Rao
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1 Answers1

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$$Z(x,t)=Ae^{pt}\sin(px)$$ $$\frac{\partial Z}{\partial t}=pZ$$ $$\frac{\partial^2 Z}{\partial x^2}=-p^2Z$$ $$\left(\frac{\partial Z}{\partial t}\right)^2=-Z\frac{\partial^2 Z}{\partial x^2}$$

JJacquelin
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  • Thank You Jacquelin, I have a query, as there are 2 arbitrary constants so the order should be 1 . But after forming Differential equation order comes out to be 2. Can you explain on this. – K S Rao May 22 '18 at 00:16
  • This is a PDE, not an ODE. Why the order should be 1 ? This is not a requirement included in the wording of the question. The solutions of a linear second order PDE are the linear combinations of two independent functions, not of only two constants. Moreover, the proposed PDE is not linear. As requested, $Z=Ae^{pt}\sin(px)$ is a solution of $\left(\frac{\partial Z}{\partial t}\right)^2=-Z\frac{\partial^2 Z}{\partial x^2}$ . But not a unique solution of course. – JJacquelin May 22 '18 at 06:35
  • Thank you Jacquelin. 2 arbitrary constants = 2 independent variables , so the solution should be unique i.e. 1 pde is only possible. Order=no. of arbitrary constants–1, so the order should be 1. This is my understanding. Please can you explain on this. – K S Rao May 22 '18 at 09:08
  • The solution of any PDE is never unique if no boundary condition is specified. I suppose that there is an ambiguity in the wording of your question. – JJacquelin May 22 '18 at 10:28
  • In your question you specify nothing about order of PDE and nothing about unicity. So, my answer is correct since $Z(x,t)=Ae^{pt}\sin(px)$ is a solution of $\left(\frac{\partial Z}{\partial t}\right)^2=-Z\frac{\partial^2 Z}{\partial x^2}$ as expected. – JJacquelin May 22 '18 at 10:33
  • Thank you Jacquelin fir your efforts. – K S Rao May 22 '18 at 11:51
  • I was just having some doubt . I will come back to you once again with another problem. Thank you Jacquelin once again – K S Rao May 22 '18 at 11:52