Is this a differential equation?

[johann1301]

My textbook says that:

"A differential equation contains both the function and the derivative of the function"

and at the same time claims that y'=3x² is a differential equation.

How can this be? The original function isn't part of the equation in this case?

 

[Simon Bridge]

Yes it is. y'=3x² is the same as y'+0y=3x² (the coefficient of y is zero) and also the same as:
xy'=3y

But I think you need to take the textbook a little less seriously.
What they mean is that if g(x)=f(x,y,y',y''...) then g is a DE.
As you learn more the definition will get expanded to include more cases.

 

[HallsofIvy]

It is a particularly easy differential equation- y is simply the anti-derivative of [tex]3x^2[/tex]. I would say the statement "A differential equation contains both the function and the derivative of the function" is at best misleading. The derivative of the function must appear explicitly in the equation. The function itself does not have to be explicitly in the equation.

 

[johann1301]

Thanks!

 

[blue_raver22]

Yes, y'=3x^2 is indeed a differential equation. This may seem confusing at first, but let me explain. A differential equation is an equation that relates a function to its derivative(s). In this case, the function is y, and its derivative is y'. So, the equation y'=3x^2 is saying that the derivative of y is equal to 3x^2. This means that y is the function whose derivative is 3x^2. In other words, y is the solution to this differential equation. So, while the original function y may not be explicitly written in the equation, it is still an important part of the overall problem and solution.