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