Linearizing Non-Linear ODEs: How to Transform Equations for Easier Solving?

[AxiomOfChoice]

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

<br /> e^{y&#039;&#039; + y} = 12<br />

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

<br /> y&#039;&#039; + y = \log 12<br />

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?

 

[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