Is a nth order ODE considered linear if n=2?

[Saladsamurai]

I am reading through my Diff Eqs Text and I follow most of the lingo. However I am just a tad confused by the statement:

An nth order ODE is said to be linear if F is linear in y,y',...y^(n)

Then it gives the example:

$$a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)..+a_0(x)y=g(x)$$

It then says: 'On the left-hand side of the above equation the dependent variable y and all of its derivatives, y,y',y'',...y$^n$ are of the first degree.

Clearly I missed something in Calc. If n=2, I have: $$\frac{d^2y}{dx^2}$$

Why is this linear if n=2?

Thanks,
Casey

 

[cepheid]

You're confusing the *order* of the derivative with its *degree* in the equation.

y'' + y = 0

is a second-order linear equation. Second-order because the highest order derivative in the equation is a second dervative. Linear because the equation itself is linear in both y'' and y.

In contrast:

(y'')^2 + ay^3 = 0

is a second-order NON-linear D.E. Second-order for the same reason. NON-linear because it is QUADRATIC in y'' and CUBIC in y. I hope this clears things up.

 

[Saladsamurai]

You're confusing the *order* of the derivative with its *degree* in the equation.

y'' + y = 0

is a second-order linear equation. Second-order because the highest order derivative in the equation is a second dervative. Linear because the equation itself is linear in both y'' and y.

In contrast:

(y'')^2 + ay^3 = 0

is a second-order NON-linear D.E. Second-order for the same reason. NON-linear because it is QUADRATIC in y'' and CUBIC in y. I hope this clears things up.

I think it does. y' just means "the 1st derivative" and similarly for y" however if either one y' or y" or y for that matter were raised to any power above 1, the DE would no longer be linear.

Thanks!

 

[HallsofIvy]

Yes, that is correct. Also note that other "non-linear" functions of the dependent variable, y or its derivatives, such as sin(y) or exp(y"), would make the equation non-linear.

 

[Saladsamurai]

Yes, that is correct. Also note that other "non-linear" functions of the dependent variable, y or its derivatives, such as sin(y) or exp(y"), would make the equation non-linear.

Great, thanks Halls and cepheid. Hey also, I know that the right-hand side can be equal to 0 or a function of the independent variable; what about a constant?

Like $\frac{d^2y}{dx^2}-\frac{dy}{dx}+6y=7$ ?