Question

Explain why both y = x-3 and y = x+2 can both be considered solutions to the differential equation dy/dx - 1 = 0?

Answer 1

See below.

Explanation:

A given function `g(x)` is a solution for the differential equation

`y'-1=0` if `g'-1=0` Now substituting `g(x) = x+C` into the differential equation we have

`d/(dx)(x+C)-1 = 1-1=0` so the solution

`y = x + C` includes `x-3` and `x+2` because `C` represents any constan in `RR`.

Answer 2

An alternative approach is to work backwards from the Differential Equation and provide the General Solution.

We have:

`dy/dx - 1 = 0`

This is a First Order separable DE, and we can write in the form:

`dy/dx = 1`

And "separating the variables" gives us:

`int y \ dx = int 1 \ dx`

Integrating we get the General Solution :

`y = x + C`

where `C` is an Arbitrary constant. Thus

`y=x-3 => C=-3`
`y=x+2 => C = -2`

Are both solutions, as is for example:

`y=x+3 \ \ => C=3`
`y=x-90 => C=-90` etc

Given an initial condition we can provide a value for the Arbitrary constant and supply a unique solution or a Particular Solution .