Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations).
indicate which pairs of functions satisfy this system. it will take some time to make all of the calculations.
y_1' = y_1 -2 y_2 \qquad y_2' = 3y_1 - 4 y_2
a. y_1 = \sin(x) +\cos(x) \qquad y_2 = \cos(x) - \sin(x)
b. y_1 = \sin(x) \qquad y_2 = \cos(x)
c. y_1 = \cos(x) \qquad y_2 = -\sin(x)
d. y_1 = e^{-x} \qquad y_2=e^{-x}
e. y_1 = e^x \qquad y_2=e^x
f. y_1 = e^{4x} \qquad y_2 = e^{4x}
g. y_1 = 2e^{-2x} \qquad y_2 = 3e^{-2x}
as you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. it greatly if we study the structure of the family of solutions to the equations. then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions.