2nd order nonlinear non-seperable equation

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The discussion centers on solving the second-order nonlinear non-separable differential equation: d2x/dt2 + (x2 - a) dx/dt + (x2 - b)x = 0. The user successfully reduces it to a first-order equation using the substitution y = dx/dt, resulting in dy/dx = (a-x2) + [(b-x2)x]/y = 0. However, they struggle with finding a solution or equilibrium states. Suggestions include exploring constant solutions and considering a series solution approach.

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phygeek
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I've recently been trying to solve the following equation:

d2x/dt2 + (x2 - a) dx/dt + (x2 - b)x = 0

I've reduced it to a first order equation by a simple substitution of y = dx/dt to obtain:

dy/dx = (a-x2) + [(b-x2)x]/y = 0

However I cannot figure out how to solve this equation. Is it possible? If not can I at least find equilibrium states?
 
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dy/dx time y comes out dy/dx times dx/dy = second derivative of x w.r.t t ?

Anyway, I see some constant solutions... Check and see. Any initial conditions on the problem?
 
Welcome to physics forum phygeek. Why don't you try a series solution.
 

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