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Answer:
Step-by-step explanation:
Assuming this complete problem: "In this problem,
y = c1ex + c2e−x
is a two-parameter family of solutions of the second-order DE
y'' − y = 0.
Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.
y(0) = 1, y'(0)= 3"
Solution to the problem
For this case we have a homogenous, linear differential equation with order 2, and with the general form:
Where
And we can rewrite the differential equation in terms like this:
And applying the second derivate we got:
We can take common factor and we got:
And for this case the two only possibel solutions are
And the general solution for this case is given by:
Replacing the roots that we found we got:
Now we can find the derivates for this last espression
From the initial conditions we have this:
(1)
(2)
If we add equations (1) and (2) we got:
And solving for we got:
So then our general solution is given by: