1) Solve :

Know : is the solution.

2 ) Solve :

Know :

is the solution.

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- Dec 17th 2009, 01:43 AMdhiabDifferential equations know one solution
1) Solve :

Know : is the solution.

2 ) Solve :

Know :

is the solution. - Dec 17th 2009, 05:16 AMKrizalid
Ricatti's equations. There's no much here. What have you done?

- Dec 17th 2009, 08:23 AMshawsend
If we know a particular solution to the Riccati equation:

then we can obtain the general solution by making the substitution:

and obtain an easier first order equation:

Now, when I do that for the first one, I obtain the general solution for this equation:

.

Now, the interesting part I think, is back-substituting that solution into the original Riccati and verifying that it satisfies it. Forgive me if I'm a little lazy and just used Mathematica:

Code:`In[61]:=`

y[x_] := Sin[x] + 1/(c1*Exp[Cos[x]] -

Exp[Cos[x]]*Integrate[Exp[-Cos[x]],

x]);

FullSimplify[D[y[x], x] + Sin[x]*y[x] -

y[x]^2]

Out[62]= Cos[x]