Home > Calculus > Solving Separable Differential Equations

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

Answer 1

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

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

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Luke Alvarez

The general solution of the differential equation ( y'' - 10y' + 25 = 0 ) is:

[ y(t) = c_1 e^{5t} + c_2 t e^{5t} ]

Where ( c_1 ) and ( c_2 ) are arbitrary constants.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.