Relationship between second order odes and pdes

[tiredryan]

Hello.

I took a class on ODEs and learned about solving second order homologous equations by writing down the characteristic equation.
http://www.sosmath.com/diffeq/second/constantcof/constantcof.html

I am now learning about PDEs on my own and I came across parabolic, hyperbolic, and elliptical operators. Is there a relationship between the characteristic equations from the 2nd order ODEs' characteristic equations and these new PDE operators or should I think of them as two separate ideas? The resources I have come across do not relate it back to ODEs.

Thanks.

 

[lippyka]

Yes, there is a relationship between the characteristic equations from ODEs and the new PDE operators. The PDEs are basically extensions of the ODEs, so the same concepts can be applied to them as well. The main difference is that the ODEs are one-dimensional and the PDEs are multi-dimensional. However, the same principles of linearity and homogeneity can still be applied. For example, in the case of parabolic equations, the equation can be reduced to a second order homogeneous ODE by using a separation of variables technique. The same principle holds for hyperbolic equations and elliptic equations. In each case, the PDE can be reduced to an ODE, and then the same techniques used to solve the ODEs can be applied to the PDEs.