The given differential equation is a linear homogeneous equation with constant coefficients. The characteristic equation is obtained by replacing y with e^(rt) in the differential equation, which gives:
rⁿe^(rt) + 5re^(rt) + 4e^(rt) = 0
Simplifying the above equation, we get:
rⁿ + 5r + 4 = 0
The roots of the above equation are r = -1, -4. Therefore, the general solution of the differential equation is given by:
y(t) = c₁e^(-t) + c₂e^(-4t) + y_p(t)
where c₁ and c₂ are constants of integration and y_p(t) is a particular solution of the non-homogeneous part of the differential equation.
To find the particular solution, we assume that y_p(t) has the form:
y_p(t) = Ae^(-3t)
where A is a constant to be determined. Substituting this into the differential equation, we get:
Ae^(-3t) - 15Ae^(-3t) + 4Ae^(-3t) = 10e^(-3t)
Simplifying the above equation, we get:
-10Ae^(-3t) = 10e^(-3t)
Therefore, A = -1. Substituting this value into our particular solution, we get:
y_p(t) = -e^(-3t)
Therefore, the general solution of the given differential equation is:
y(t) = c₁e^(-t) + c₂e^(-4t) - e^(-3t)
where c₁ and c₂ are constants of integration.
I hope this helps!