What is the general solution of the differential equation # y'' - 10y' +25 = 0#?

1 Answer
Steve M
Jun 20, 2017

# y = Axe^(5x) + Be^(5x) #

Explanation:

We have:

# y'' - 10y' +25 = 0# ..... [A]

This is a Second order linear Homogeneous Differentiation Equation with constant coefficients. The standard approach is to find a solution, #y_c# of the homogeneous equation by looking at the Auxiliary Equation, which is the polynomial equation with the coefficients of the derivatives

Complimentary Function

The associated Auxiliary equation is:

# m^2-10m+25 = 0#
# (m-5)^2 #

Which has repeated real solutions #m=5#

Thus the solution of the homogeneous equation is:

# y_c = (Ax+B)e^(5x)#
# \ \ \ = Axe^(5x) + Be^(5x) #

Confirming the quoted solution

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