Why is y'=3x² considered 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 $$3x^2$$. 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!