Question

What is a solution to the differential equation `(dy/dx) +5y = 9`?

Answer 1

`y = 9/5 + Be^(-5x)`

Explanation:

This is a First Order Separable Differential Equation.

We can isolate the variables as follows;

`dy/dx + 5y = 9`
`:. dy/dx = 9 - 5y`

Separating the variable gives us:

`:. int 1/(9 - 5y)dy = int dx`
`:. -int 1/(5y-9)dy = int dx`

Integrating gives us:

`-1/5 ln|5y-9| = x + C`
`:. ln|5y-9| = -5x -5C`
`:. 5y-9 = e^(-5x -5C)`
`:. 5y-9 = e^(-5x)e^( -5C)`
`:. 5y-9 = Ae^(-5x)`
`:. 5y = 9 + A/5e^(-5x)`
`:. y = 9/5 + Be^(-5x)`

Verification of Solution:

`y = 9/5 + Be^(-5x)`
`y = -5Be^(-5x)`

So, `dy/dx + 5y = -5Be^(-5x) + 5{9/5 + Be^(-5x)}`
`:. dy/dx + 5y = -5Be^(-5x) + 9 + 5Be^(-5x)`
`:. dy/dx + 5y = 9` QED