Question

How to check work for First Order Linear Differential Equation?

So for example, I calculated what y is from the equation. According to our book, I should take the derivative of that answer to check work. But how in the world do you take a derivative with a +C in there. The C will stay there afterwards, so there is almost no way to check it? Or did I misunderstand something?

Answer 1

The presence of `C` should not make a difference!

Explanation:

Let me explain with an example. Consider the linear first order ODE

`dy/dx + y = x^2`

By using standard methods you can arrive at the solution

`y(x) = 2-2x+x^2+color(red)Ce^-x`

Let us see how you can check this solution by taking the derivative. The derivative is easily seen to be

`dy/dx = d/dx(2-2x+x^2+color(red)Ce^-x)`
`qquad = -2+2x-color(red)Ce^-x`

Both the solution and its derivative contain a term which has the arbitrary constant of integration `color(red)C`. When you substitute these on the left hand side of the equation, you get

`"LHS" = dy/dx+x`
`qquad = [-2+2x-color(red)Ce^-x]+[ 2-2x+x^2+color(red)Ce^-x]`

As you can easily see, the right hand side evaluates to `x^2` - which shows that the solution we have arrived at is the correct one. The terms containing the arbitrary constant `color(red)C` cancel out!