How to Approach Solving a Nonlinear Second Order ODE with a Quadratic Term?

  • #1
Safinaz
260
8
Homework Statement
How to solvebthis second-order ODE:
Relevant Equations
##
\frac{\partial^2 x}{ \partial t^2} + b \frac{\partial x}{ \partial t} + C x - D x^2 =0
##

Or:

##
\ddot{x} + b \dot{x} + C x - D x^2 =0
##
Where

## b, C, D ## are constants.
I know how to solve similar ODEs like

##
\frac{\partial^2 x}{ \partial t^2} + b \frac{\partial x}{ \partial t} + C x =0
##

Where one can let ## x = e^{rt}##, and the equation becomes
##
r^2 + b r + C =0
##

Which can be solved as a quadratic equation.

But now the problem is that there is ##x^2## term, so if one used that substitution, we left by:
##
r^2 + b r + C + D e^{rt} =0
##

So any help to find the solution of the ODE
 
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  • #2
Safinaz said:
Where one can let x=ert, and the equation becomes
r2+br+C=0

Which can be solved as a quadratic equation.
You have got general solution of homogeneous differential equation. Then you have to find a particular solution to add that for inhomogeneous differential equation with x^2 term. Have you investigated x=constant ?
 
Last edited:
  • #3
anuttarasammyak said:
You have got homogeneous general solution. Then you have to find a special solution to add that. Have you investigated x=constant ?
The OP's equation is a non linear ODE?
 
  • #4
erobz said:
The OP's equation is a non linear ODE?
My bad. Thanks. By choosing sign of constants, the equation is interpreted as oscillation of a body in a viscous medium with harmonic if D=0 and inharmonic with D potential. x = 0 is stable, x= C/D is unstable point for small oscillation around.
 
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  • #5
Safinaz said:
##\frac{\partial^2 x}{ \partial t^2} + b \frac{\partial x}{ \partial t} + C x - D x^2 =0
##
Or:
##
\ddot{x} + b \dot{x} + C x - D x^2 =0
##
The second version of your DE, using the notation with dots, suggests that x is a function of t alone. In that case the first version of the DE should be written without partials.
Like so:
##\frac{d^2 x}{dt^2} + B\frac{dx}{dt} + Cx - Dx^2 = 0##
Also, since C and D are uppercase, B should probably be uppercase as well.
erobz said:
The OP's equation is a non linear ODE?
Yes. I'm sure your question was rhetorical.
 
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