Undetermined Coefficients Method for Solving ODEs

[Unassuming]

Homework Statement

(I am using b, for beta)

Let

$$f(t)=ce^{-\beta t}$$

with b a fixed number and c in R is arbitrary but given.

Write the general solution of

$$x'=-x+f(t)=-x+ce^{-\beta t}$$

Hint: Use undetermined coeff. and consider a particular solution of the form

$$x_p=\alpha e^{-\beta t}$$

and determine $$\alpha$$

Homework Equations

The Attempt at a Solution

I have solved for

$$\alpha = \frac{c}{-\beta +1}$$

 

[tiny-tim]

Write the general solution of

$$x'=-x+f(t)=-x+ce^{-\beta t}$$
…
I have solved for

$$\alpha = \frac{c}{-\beta +1}$$

Hi Unassuming!

(have an alpha: α and a beta: β )

What's the difficulty?

You have found a particular solution, (c/(1-β))e^-βt.

Now just find the complementary solution, ie the solution to x' = -x, and add it.