How Do You Solve a Second-Order Linear PDE with Given Boundary Conditions?

[pablo4429]

Finding basic solutions to a PDE??

So the problem is:
x_o=0
\varphi'' + 4\varphi' + \lambda\varphi=0

which satisfies \varphi(0)=3 and \varphi'(0)=-1

I really don't even know where to start, I think its like an ODE right where we assume a solution, usually sin or an exponential and plug it in for each psi and its derivatives, find roots and plud back into a general solution and use BC to find constants. In the text though, they give psi as a linear combo of psi 1 and psi 2 with some coefficients in front. The answer they give is an exponential multiplied by a sin term and a cos term for psi 1 and an exponential multiplied by a sin term.
thanks for any help all

 

[gato_]

the equation is linear with constant coefficients. Try
\phi=Ae^{st}
which will give you a condition on s, for which you get two solutions, s_{1},s_{2}. Then plug
\phi=Ae^{s_{1}t}+Be^{s_{2}t}
into the boundary conditions to get the coefficients