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Homework Statement
Find a solution [itex] \bf{\phi} [/itex] of the system
$$y'_1(t)=y_1(t)+y_2(t)+f(t)$$
$$y'_2(t)=y_1(t)+y_2(t)$$
where f(t) is a continuous function
$$\bf{\phi} (0)=(0,0)$$
Homework Equations
A hint was given to define ##v(t)=y_1(t)+y_2(t)##
The Attempt at a Solution
Using the suggested substitution, the system became
$$v'(t)=v(t)+y'_2(t)+f(t)$$
$$v'(t)=v(t)+y'_1(t)$$
I then added these together to get
$$v'(t)-2v(t)=f(t)$$ where $$v'(t) = y'_1(t)+y'_2(t)$$
This was solved using an integrating factor to get
$$v(t)=e^{2t} \int_0^t e^{-2t}f(t)dt$$
but then from here I have no idea where to go, if this was even the proper path to begin with.
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