Nth Order Differential Equations

[ahaanomegas]

We know that there are a few forms for 1st order differential equations. Second-order differential equations have an extra term with an $x$ in it. My conjecture is that third-order differential equations have another extra term with an $x^2$ in it. A friend of mine agrees with this. Is this true and can we following the same pattern for higher order differential equations? If so, is there a proof to this?

Thanks in advance! :)

 

[JJacquelin]

We know that there are a few forms for 1st order differential equations. Second-order differential equations have an extra term with an $x$ in it. My conjecture is that third-order differential equations have another extra term with an $x^2$ in it. A friend of mine agrees with this. Is this true and can we following the same pattern for higher order differential equations? If so, is there a proof to this?

Thanks in advance! :)

What do you mean in writing "We know that there are a few forms for 1st order differential equations" ?
This seens rather ambiguous. What kind of ODE are you talking about ?
The general solution of a first order ODE is on the form f(c,x) where c is an arbitrary constant. The form c*f(x) is not the general case : it is only true in the case of linear ODE.
The general solution of a second order ODE is on the form f(c1,c2;x) where c1 and c2 are arbitrary constants. The form c1*f1(x)+c2*f2(x) is not the general case : it is only true in the case of linear ODE.
The general solution of a third order ODE is on the form f(c1,c2,c3;x) where c1, c2, c3 are arbitrary constants.
etc.

 

[HallsofIvy]

We know that there are a few forms for 1st order differential equations. Second-order differential equations have an extra term with an $x$ in it. My conjecture is that third-order differential equations have another extra term with an $x^2$ in it. A friend of mine agrees with this. Is this true and can we following the same pattern for higher order differential equations? If so, is there a proof to this?

Thanks in advance! :)

The most general way of writing a first order differential equation is f(x, y, y')= 0.
The most general way of writing a second order differential equation is f(x, y, y', y'')= 0.
The most general way of writing a differential equation of order n is $f(x, y, y', y'', ..., y^{(n)})= 0$

I have no idea what you mean by "an extra term with and x in it" nor "another extra term with an $x^2$ in it". The second order equation y''= y has no explicit "x" at all.