- #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
##
\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