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

Jun 20, 2017

$y = A x {e}^{5 x} + B {e}^{5 x}$

#### Explanation:

We have:

$y ' ' - 10 y ' + 25 = 0$ ..... [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:

${m}^{2} - 10 m + 25 = 0$
${\left(m - 5\right)}^{2}$

Which has repeated real solutions $m = 5$

Thus the solution of the homogeneous equation is:

${y}_{c} = \left(A x + B\right) {e}^{5 x}$
$\setminus \setminus \setminus = A x {e}^{5 x} + B {e}^{5 x}$

Confirming the quoted solution