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If function $y=x^2$, then the derivative of $y$ is $2x$. We write the derivative as either $f'(x)=2x$ or $\frac{\text dy}{\text dx}=2x$.

In this case $\frac{\text dy}{\text dx}=2x$ is also a differential equation.

But if we have the function $y=ax^2+bx+c$

then in this case we will have $\frac{\text dy}{\text dx}= 2ax+b$.

Is $\frac{\text dy}{\text dx}= 2ax+b$ also considered as a differential equation or we will have to remove all arbitrary constants from the function to call it a differential equation?

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1 Answer

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You can view the derivative operator $D(f)=f'=\frac{\mathrm d}{\mathrm dx}f$ as a function which produces a new function, it's derivative. So the derivative of a function is a new function which gives some information about the old function.

A differential equation is an equation with a function and its derivatives. Where you try to solve for the unknown function.

A first order linear differential equation looks like:$$f'(x)+a(x)f(x)=R(x)$$So the examples you gave were differential equations with $a(x)=0$.
But differential equations can be much more complicated then a derivative. A simple example is:$$f'(x)=f(x)$$ With a solution $e^x$. Second order linear differential equations look like:$$f''(x)+a(x)f'(x)+f(x)=R(x)$$But even a monster like:$$f'''(x)\cdot \sin(f(x))=e^x$$ Is a differential equation.

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