# Riccati differential equations

by mcmaster1987
Tags: differential, equations, riccati
 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.
 P: 1,666 When I make the usual substitution: $$u=\frac{y'}{Ry}$$ with $R=-1$ in your case (see info about Riccati equation), I get: $$y''+t^4y=0$$ Suppose you had to come up with an analytic expression for the eqn. in y. What would you do?
P: 2
 Quote by jackmell When I make the usual substitution: $$u=\frac{y'}{Ry}$$ with $R=-1$ in your case (see info about Riccati equation), I get: $$y''+t^4y=0$$ Suppose you had to come up with an analytic expression for the eqn. in y. What would you do?

why do you make u=y'/Ry and why you take R=-1

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.

 P: 1,666 Riccati differential equations Ok, my bad. It should be as you said and that's called the parabolic cylinder differential equation: $$y''+x^2y=0$$ 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: $$y(x)=\sum_{n=0}^{\infty}a_nx^n$$ Then the solution to the original DE is: $$u(x)=-\frac{\frac{d}{dx} \sum_{n=0}^{\infty}a_nx^n}{\sum_{n=0}^{\infty}a_nx^n}$$ Nothing wrong with that is there?

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