The nonlinear differential equation
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(1) |
one can convert it to a second order homogeneous linear differential equation with non-constant coefficients.
If one can find a particular solution
, then one can easily verify that the substitution
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(2) |
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(3) |
Example. The Riccati equation
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(4) |
We substitute
to (4), getting
For solving this first order equation we can put
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(5) |
After separation of variables and integrating, we obtain from here a solution
Separating the variables yields
and integrating:
Thus we have
whence the general solution of the Riccati equation (4) is
It can be proved that if one knows three different solutions of Riccati equation (1), then any other solution may be expressed as a rational function of the three known solutions.
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