A oneparameter family of solutions to the DE z z 3 is z c

A one-parameter family of solutions to the DE z\' + z = 3 is z = ce^-x + 3. Does any solution curve pass through the point (2, 5)? Show work to support your answer. Does any solution curve pass through the point (0, 0)? Show work to support your answer.

Solution

a. z=ce^{-x}+3

Substituting (2,5) in equation to get C we get

5=ce^{-2}+3

c=2e^2

Hence there is a solution curve passing through (2,5)

b.

Now setting (0,0) in general solution gives

0=ce^{-0}+3

c=-3

Hence again we are able to solve for c hence there is a solution curve passing through (0,0)