Question

How would I know if a differential equation is linear or non-linear?

Answer 1

http://answers.yahoo.com/question/index?qid=20090202172952AAHlxqr

Answer 2

help? by giving some other examples?

Answer 3

http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx

Answer 4

A differential equation has the form \(y^{(n)}=f(y^{(n-1)},y^{(n-2)},...,y'',y',y,g(x))\).
(So in other words, you're given a function of some-order derivative of \(y\) in terms of its lower-order derivatives of \(y\) and the variable \(x\). The reason I use \(g(x)\) is because you can have \(g(x)=x^2\), for instance, and it won't change the linearity of the DE.)

The equation is linear if you can write
\[y^{(n)}=c_1y^{(n-1)}+c_2y^{(n-2)}+\cdots+c_{n-2}y''+c_{n-1}y'+c_ny+c_{n+1}g(x)\]
Basically, a non-linear equation would include a power of y, such as \(y^2\), or y multiplied by one of its derivatives, like \(y~y^{(4)}\). The inclusion of functions of y also make an equation non-linear, such as \(\sin y\) or \(e^y\).

Some examples of linear equations are
\[y''+2y'-y=x\\
\pi y'=\sin x\\
\sum_{n=1}^{100}y^{(n)}+y=\frac{\left((\sqrt x)^{\sqrt x}\right)^{\sqrt x}}{\sin(\sin(\sin x))}\]
Some examples of non-linear equations are
\[\left(y'\right)^2-y^5=0\\
y^8y''=\frac{18x}{\ln y}\\
y^y\sqrt{y'}=e\]