To solve this second-order differential equation, we can first find the characteristic equation by substituting y = e^(rt) into the equation:
Y'' = r^2e^(rt)y' = re^(rt)
Therefore, the equation becomes:
r^2e^(rt) - re^(rt) - 6e^(rt) = 0(e^(rt))(r^2 - r - 6) = 0
The characteristic equation is then:
r^2 - r - 6 = 0
Now, we can solve this quadratic equation for r by factoring or by using the quadratic formula:
r^2 - r - 6 = 0(r - 3)(r + 2) = 0
This gives us two roots: r1 = 3 and r2 = -2
Therefore, the general solution to the differential equation is:
y(t) = C1e^(3t) + C2e^(-2t)
where C1 and C2 are arbitrary constants that can be determined from initial conditions if they are provided.
To solve this second-order differential equation, we can first find the characteristic equation by substituting y = e^(rt) into the equation:
Y'' = r^2e^(rt)
y' = re^(rt)
Therefore, the equation becomes:
r^2e^(rt) - re^(rt) - 6e^(rt) = 0
(e^(rt))(r^2 - r - 6) = 0
The characteristic equation is then:
r^2 - r - 6 = 0
Now, we can solve this quadratic equation for r by factoring or by using the quadratic formula:
r^2 - r - 6 = 0
(r - 3)(r + 2) = 0
This gives us two roots: r1 = 3 and r2 = -2
Therefore, the general solution to the differential equation is:
y(t) = C1e^(3t) + C2e^(-2t)
where C1 and C2 are arbitrary constants that can be determined from initial conditions if they are provided.