- #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