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Riccati differential equations 
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#1
Oct1410, 11:39 AM

P: 2

riccati differential equations
 how to find general solution of this question du/dt=u^2+t^2 please say me i work hard but i do nat know this form of riccati equation. i know when special solution is given however there is no special soltion such that u=u1(t) in this question. 


#2
Oct1410, 06:23 PM

P: 1,666

When I make the usual substitution:
[tex]u=\frac{y'}{Ry}[/tex] with [itex]R=1[/itex] in your case (see info about Riccati equation), I get: [tex]y''+t^4y=0[/tex] Suppose you had to come up with an analytic expression for the eqn. in y. What would you do? 


#3
Oct1510, 11:03 AM

P: 2

why do you make u=y'/Ry and why you take R=1 please tell me. and i think, you have made an error process. because you found an equation y''+yt^4=0 but i found y''+yt^2=0. and then i used second order linear differential equation thecniques. After that i found y=C1e^it+C2e^(it). i think this is not true. Thank you for your interest my question. 


#4
Oct1510, 02:15 PM

P: 1,666

Riccati differential equations
Ok, my bad. It should be as you said and that's called the parabolic cylinder differential equation:
[tex]y''+x^2y=0[/tex] But that's not solved using ordinary techniques. You could however, use power series and that's what I was referring to above. Say you get it in the form: [tex]y(x)=\sum_{n=0}^{\infty}a_nx^n[/tex] Then the solution to the original DE is: [tex]u(x)=\frac{\frac{d}{dx} \sum_{n=0}^{\infty}a_nx^n}{\sum_{n=0}^{\infty}a_nx^n}[/tex] Nothing wrong with that is there? 


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