Differential Equations

[SoSolid]

I have a 3rd order differential equation and want to reduce it to a linear first oder differential equation. Does anybody know how to go about this?

 

[danielmp007]

http://letmegooglethatforyou.com/?q...to+a+linear+first+order+differential+equation

 

[SoSolid]

Thanks, I already tried that. It does not help.

 

[Duff-Man]

You can reduce any (linear) n-th order ordinary differential equation into n coupled first order ODEs simply enough:

Your third order ODE is:

a.x''' + b.x'' + c.x' + dx + e = 0

Introduce variables q, r:

y = x'
z = x'' (= y')

Then your equation becomes:

a.z' + b.z + c.y + d.x + e = 0
y - x' = 0
z - y' = 0

... three coupled first-order ODEs You can solve these problems easy enough.
Incidentally, this procedure is the basis of solving partial differential equations using (traditional) numerical methods. You break down the PDE into a massive system of linearlised ODEs, and solve using Runge-Kutta or similar.

 

[SoSolid]

You can reduce any (linear) n-th order ordinary differential equation into n coupled first order ODEs simply enough:

Your third order ODE is:

a.x''' + b.x'' + c.x' + dx + e = 0

Introduce variables q, r:

y = x'
z = x'' (= y')

Then your equation becomes:

a.z' + b.z + c.y + d.x + e = 0
y - x' = 0
z - y' = 0

... three coupled first-order ODEs You can solve these problems easy enough.
Incidentally, this procedure is the basis of solving partial differential equations using (traditional) numerical methods. You break down the PDE into a massive system of linearlised ODEs, and solve using Runge-Kutta or similar.

Thanks, but the equation is like this

((2d^3) F /dn^3 )+ F ((d^2)F/dn^2) = 0 I am confused by the second term as it is F and we are differentiating with respect to F.

 

[Duff-Man]

Yeah, that makes it a nonlinear equation.

Anyway, you can apply the same procedure:

You have:

2.F''' + F.F'' = 0

set:
p = F''
q = F'

To get:

2p' + F.q' = 0
q - F' = 0
p - q' = 0

You will need an iterative procedure to solve this NL equation (ie start with an initial quess for one of the variables, probably F. Solve using this guess value, and then update the guess value of F. Repeat until convergence).

 

[SoSolid]

Yeah, that makes it a nonlinear equation.

Anyway, you can apply the same procedure:

You have:

2.F''' + F.F'' = 0

set:
p = F''
q = F'

To get:

2p' + F.q' = 0
q - F' = 0
p - q' = 0

You will need an iterative procedure to solve this NL equation (ie start with an initial quess for one of the variables, probably F. Solve using this guess value, and then update the guess value of F. Repeat until convergence).

Yup I am going to write a runge kutta algorithm in matlab.