Final answer:
To solve the third-order linear homogeneous differential equation, find the characteristic equation, solve it for its roots, and then express the general solution in terms of these roots and exponential functions with arbitrary constants.
Step-by-step explanation:
The differential equation given is a third-order linear homogeneous differential equation:
y"' + y" + 8y' - 10y = 0
.
To solve this type of equation, we first need to find the characteristic equation which is done by substituting y with ert where r is the unknown constant we need to solve for. After substitution, our differential equation becomes:
r
3
+ r
2
+ 8r - 10 = 0
.
Now, we need to find the roots of this characteristic equation. This is often done by trial and error unless the roots can be easily observed, or by using methods such as synthetic division or the Rational Root Theorem. After finding the roots of the equation, say r1, r2, and r3, the general solution to the differential equation can be expressed as:
y(t) = C
1
e
r
1
t
+ C
2
e
r
2
t
+ C
3
e
r
3
t
.
Where C1, C2, and C3 are constants determined by the initial conditions of the specific problem.