Originally Posted by
dhiab 1) Solve :
Know :
is the solution.
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]
I'm sure you could do the second one now.