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

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The discussion focuses on the linearization of non-linear ordinary differential equations (ODEs), specifically examining the equation e^{y'' + y} = 12. It is established that this equation is not linear but can be transformed into a linear form by applying logarithmic operations, resulting in y'' + y = log 12. The conversation also highlights the importance of transformations, such as finding integrating factors for first-order equations and changing variables to linearize inverse-linear equations like dy/dx = 1/(x-y(x)).

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  • Understanding of ordinary differential equations (ODEs)
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So, this is probably really easy, but it's been bugging me...is the following differential equation linear?

[tex] e^{y'' + y} = 12[/tex]

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

[tex] y'' + y = \log 12[/tex]

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?
 
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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
 

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