Distinguishing between linear and non-linear differential equations

I am working on a few problems from Dennis Zill's book on Differential equations and in te exercise below I am asked to say if the differential equation is linear or non-linear and its order:

1 - 2nd order, linear

2 - 3rd order, ?

3 - 4th order, linear

4 - 2nd order, non-linear

5 - 2nd order, non-linear

6 - 2nd order, non-linear

7 - 3rd order, linear

8 - 2nd order, ?

Can someone confirm that? I am really confused about a differential equation being linear or nonlinear.

• $2.$ and $8.$ are nonlinear. – Moo Jun 25 '17 at 21:23

An ordinary differential equation is linear if it can be written in the form $$L(y(x))=\left[A_n(x)\frac{d^n}{dx^n}+A_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}+ \cdots +A_1(x)\frac{d}{dx}+A_0(x)\right]y(x)=f(x)$$
(this guarantees that if $h(x)$ and $k(x)$ are solutions , also $\alpha h(x)+\beta k(x)$ is a solution) so, in your case , the equations 2,4,5,6,8 are not linear.