# Solving Differential Equation using method of undetermined coefficients

The problem:

A spring is stretched 6 in by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of 0.25 lb⋅s/ft and is acted on by an external force of 5cos(2t) lb. Determine the steady-state response of this system. Assume that g=32 ft/s^2.

Express your answer as a linear combination of sin(at) and cos(at).

My Solution:

I use the information to create a second order differential equation:

$$\frac{1}{4}u''+ \frac{1}{4}u'+16u=5 cos(2t)$$ With the initial contions :$$u(0)=0,u'(0)=0$$ I can solve for the homogeneous solution but I do not understand how to find the particular solution

## 1 Answer

For such a function you should try $$u_p=a\cos2t+b\sin2t.$$ Plug that into your equation and find $$a,b$$ (using that $$\cos2t$$ and $$\sin2t$$ are linearly independent).

• I got a solution, but I dont see how it can be expressed as a linear combination of sin(at) and cos(at) – Doldrums Nov 8 '18 at 2:48