1. Feb 26, 2013

### AxiomOfChoice

So, this is probably really easy, but it's been bugging me...is the following differential equation linear?

$$e^{y'' + y} = 12$$

'Cause can't you just take logarithms on both sides and get it to be

$$y'' + y = \log 12$$

I guess the question I'm trying to ask is...what operations are you allowed to take an ODE through in trying to put it in linear form?

2. Feb 26, 2013

### bigfooted

Well, the first equation is not a linear ODE, but it can be linearized easily. Most strategies for solving ODE's are based around a transformation to a form that is easily solvable. For first order equations, you usually try to transform the ODE to an exact ODE by finding an integrating factor.

Another example, this equation:

dy/dx = 1/(x-y(x))

is inverse-linear. You can linearize it if you change the dependent and independent variables x->x(y) and y(x)->y and you will get:

dy/dx = 1/(x(y) - y)
dx/dy = x(y) - y, or:
x' = x - y