Differential Equations 2nd Order Linear Constant-coefficient Problem

[jake2]

Homework Statement

solve:
160y''=160g-ky
y(0)=-200 and y'(0)=0

2. The attempt at a solution

I tried to use guess and check to solve this equation, but it didn't turn out nice at all...

y''=9.8 - (ky)/160
y''+(ky)/160 = 9.8, guess y=e^(λt), y'=λe^(λt), y''=λ^2e^(λt) : this gives

e^(λt)*(λ^2+(k/160))=9.8

Every problem that I've done has not had a term without a y'',y',and a y, so I have no idea where to go from here, or alternative methods to solving this problem.

 

[Mute]

This is a linear, second order non-homogeneous problem. The general solution to a non-homogeneous problem looks like y(t) = y_h(t) + Y_p(t), where y_h(t) is the general solution to the homogeneous problem (i.e., a problem where some function of y and its derivatives = 0, and not some constant or other function) and Y_p(t) is the particular solution for the inhomogeneous part.

When you start by trying a solution of the form y(t) = exp(λt), that's good, but it will only help you with the homogeneous problem 160 y'' + ky = 0. Solving that problem will give you the homogeneous solution. Then, can you guess what kind of particular solution you should add to that to get the full inhomogeneous solution?

 

[ehild]

Have you learned about differential equations? This is an inhomogeneous linear equation: the term without y makes it inhomogeneous. You get the solution if you solve the homogeneous equation first (that which contains y, y', y" only): 160y"+ky=0, than add a particular solution of the original equation. You get that particular solution if this equation if you assume that y"=0.

ehild

EDIT: Mute beat me...