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?

1 Answer
May 27, 2018

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


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!