Section 5.2 Problem 7: Find the general solution y'' - 8y' + 16y = 0 ​

Question

Section 5.2 Problem 7:

Find the general solution

`y'' - 8y' + 16y = 0`
​

Answer 1

`y(x)=C_1e^(4x)+C_2e^(4x)`

Explanation:

To solve a second-order homogeneous differential equation, we need to substitute each term with the auxiliary equation
`am^2+bm+c=0` where the values of
`m` are the roots:

`y''-8y'+16y=0\\\\m^2-8m+16=0\\\\(m-4)^2=0\\\\m-4=0\\\\m=4`

Since the values of
`m` are equal real roots, then the general solution is
`y(x)=C_1e^(m_1x)+C_2e^(m_1x)`

Thus, the general solution for our given differential equation is
`y(x)=C_1e^(4x)+C_2e^(4x)`