What is the General Solution of the Differential Equation y''-6y'+10y = 0?
2 Answers
We have;
y''-6y'+10y = 0
This is a Second Order Homogeneous Differential Equation which we solve as follows:
We look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.
m^2-6m+10 = 0
This quadratic does not factorise to I will solve by completing the square (you could equally use the quadratic formula)
(m-3)^2-3^2+10 = 0
:. (m-3)^2 = -1
:. m-3 = +-i
:. m-3 = 3+-i
Because this has two distinct complex solutions
y = e^(pt)(Acosqt+Bsinqt)
Where
y = e^(3t)(Acost+Bsint)
Explanation:
The general solution for this kind of differential equation (homogeneous linear with constant coefficients) is
substituting into the differential equation we have
but
with
Using de Moivre's identity
Here