How to prove (global) uniqueness of solution to linear, first order ODE?

Martin yesterday at 04:49. 1 answers, 0 views

Consider the first order linear ODE

$F(x) + A x F'(x) + B = 0$, with initial condition $F(1) = 1$. Moreover, let $A,B \neq 0$, real, and $sgn(A) = sgn(B)$.*

*... if the latter matters.

Mathematica gives me the following solution, which is easy to verify:

$F(x) = (1+B)(x^{-1/A}) - B$.

My question is: is there any elementary way or theorem to establish that the above must be the unique global solution to the given ODE? Do I have to apply the Picard–Lindelöf theorem (if that works), or is there an easier argument?

I have not much knowledge about how to solve differential equations; the question arises from an applied economics paper I'm working on.

1 Answers

Hans Engler 06/04/2013.

You can't expect "global" uniqueness (for all real $x$). The reason is that the differential equation provides no information about $F'(0)$ and thus the portions for $x > 0$ and for $x < 0$ "don't know about each other".

Here's how to prove uniqueness on $(0, \infty)$.Suppose you have two solutions $F$ and $G$ of the same problem. Set $f(x) = F(x) - G(x)$ for $x > 0$, then $Axf'(x) + f(x) = 0$ and $f(1) = 0$. Now set $g(x) = x^{1/A}f(x)$. Then you can check that $g'(x) = 0$ for $x > 0$ and $g(1) = 0$. Therefore $g(x) = 0$ for all $x >0 $ and therefore $F(x) = G(x)$ for all $x > 0$.

Here is an example for two solutions that agree for $x > 0$ but are different for $x < 0$. Consider the two functions $$ F(x) = \begin{cases} (1+B)x^{-1/A} - B \quad (x > 0)\\ B x^{-1/A} - B \quad (x < 0) \end{cases} $$ and $$ G(x) = (1+B)x^{-1/A} - B \quad (x \ne 0) \, . $$ Set $F(0) = G(0) = -B$. Both functions satisfy the differential equation and the condition $F(1) = 1$, but they differ for $x < 0$. - Download Hi-Res Songs

1 The Chainsmokers

Beach House flac

The Chainsmokers. 2018. Writer: Andrew Taggart.
2 (G)I-DLE


(G)I-DLE. 2018. Writer: Riot Music Team;Harloe.
3 Anne-Marie

Rewrite The Stars flac

Anne-Marie. 2018. Writer: Benj Pasek;Justin Paul.
4 Ariana Grande

​Thank U, Next flac

Ariana Grande. 2018. Writer: Crazy Mike;Scootie;Victoria Monét;Tayla Parx;TBHits;Ariana Grande.
5 Diplo

Close To Me flac

Diplo. 2018. Writer: Ellie Goulding;Savan Kotecha;Peter Svensson;Ilya;Swae Lee;Diplo.

Waste It On Me flac

BTS. 2018. Writer: Steve Aoki;Jeff Halavacs;Ryan Ogren;Michael Gazzo;Nate Cyphert;Sean Foreman;RM.
7 Clean Bandit

Baby flac

Clean Bandit. 2018. Writer: Jack Patterson;Kamille;Jason Evigan;Matthew Knott;Marina;Luis Fonsi.
8 Imagine Dragons

Bad Liar flac

Imagine Dragons. 2018. Writer: Jorgen Odegard;Daniel Platzman;Ben McKee;Wayne Sermon;Aja Volkman;Dan Reynolds.
9 BlackPink

Kiss And Make Up flac

BlackPink. 2018. Writer: Soke;Kny Factory;Billboard;Chelcee Grimes;Teddy Park;Marc Vincent;Dua Lipa.
10 Nicki Minaj

No Candle No Light flac

Nicki Minaj. 2018. Writer: Denisia “Blu June” Andrews;Kathryn Ostenberg;Brittany "Chi" Coney;Brian Lee;TJ Routon;Tushar Apte;ZAYN;Nicki Minaj.
11 Rita Ora

Cashmere flac

Rita Ora. 2018. Writer: Sean Douglas;Lindy Robbins.
12 Backstreet Boys

Chances flac

Backstreet Boys. 2018.
13 Brooks

Limbo flac

Brooks. 2018.
14 Rita Ora

Velvet Rope flac

Rita Ora. 2018.
15 Fitz And The Tantrums

HandClap flac

Fitz And The Tantrums. 2017. Writer: Fitz And The Tantrums;Eric Frederic;Sam Hollander.
16 Little Mix

Woman Like Me flac

Little Mix. 2018. Writer: Nicki Minaj;Steve Mac;Ed Sheeran;Jess Glynne.
17 Cher Lloyd

None Of My Business flac

Cher Lloyd. 2018. Writer: ​iamBADDLUCK;Alexsej Vlasenko;Kate Morgan;Henrik Meinke;Jonas Kalisch;Jeremy Chacon.
18 Billie Eilish

When The Party's Over flac

Billie Eilish. 2018. Writer: Billie Eilish;FINNEAS.
19 Kelly Clarkson

Never Enough flac

Kelly Clarkson. 2018. Writer: Benj Pasek;Justin Paul.
20 Lil Pump

Arms Around You flac

Lil Pump. 2018. Writer: Rio Santana;Lil Pump;Edgar Barrera;Mally Mall;Jon Fx;Skrillex;Maluma;Swae Lee;XXXTENTACION.


Popular Tags