Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Calculus and Beyond Homework Help
Maxima and minima of differential equation
Reply to thread
Message
[QUOTE="trash, post: 4864444, member: 482453"] [h2]Homework Statement [/h2] Consider the differential equation [itex]y'=x-y^2[/itex]. Find maxima, minima and critical points; show that for every solution [itex]f=f(t)[/itex] exists [itex]T\geq 0[/itex] such that [itex]f(t)< \sqrt{T}\;\forall t > T[/itex] [h2]Homework Equations[/h2] The Riccati equation: [itex]y'=a(x)y^2+b(x)y+c(x)[/itex] The Bernoulli equation: [itex]y'=a(x)^n+b(x)y[/itex] [h2]The Attempt at a Solution[/h2] [/B] I've been trying to study the differential equation given by [itex]y'=x-y^2[/itex] for a while, and I didn't was even close to a solution. Finally when I gave up I solved this equation with Wolfram Alpha, and it seems that the solution was quite intricate and doesn't seem possible to solve only with elementary functions, see [URL='http://www.wolframalpha.com/input/?i=y%27%3Dx-y%5E2&lk=4&num=1']Wolfram's solution[/URL]. Now my question is, what are the possible methods to use when I encounter a non-linear equation of like this?, is it possible to say something about it without the need of a computer?. One thing I saw is that this is similar the Riccati equation [itex]y'=a(x)y^2+b(x)y+c(x)[/itex] using [itex]c(x)=x, a(x)=-1, b(x)=0[/itex] and is possible to get a general solution if I know two other solutions: if [itex]f_1,f_2[/itex] are solutions then [itex]f=f_1+C(f_2-f_1)[/itex] is a general solution. The problem here is that I'd need to find particular solutions for the equations, something that I couldn't figure out how to do it in the general case. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Calculus and Beyond Homework Help
Maxima and minima of differential equation
Back
Top