Final answer:
The general solution to the differential equation y" - 3y' + 2y = 0 is y = C1e^t + C2e^2t, where C1 and C2 are constants determined by initial conditions.
Step-by-step explanation:
General Solution to the Differential Equation
To find the general solution to the differential equation y" - 3y' + 2y = 0, we need to look for solutions of the form y = ert, where r is a constant. Substituting this form into the differential equation, we get:
r2ert - 3rert + 2ert = 0
Factoring out ert, we have:
(r2 - 3r + 2)ert = 0
Since ert is never zero, we must have:
r2 - 3r + 2 = 0
This quadratic equation factors to:
(r - 1)(r - 2) = 0, giving us the roots r = 1 and r= 2.
The general solution is therefore:
y = C1et + C2e2t
Where C1 and C2 are arbitrary constants that can be determined by initial conditions.