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November 12th, 2017, 11:59 PM  #1 
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Constant solution
I want to find constant solutions of the differential equation $\frac{dy}{dx}=\frac{y^3}{x^2}2\frac{y}{x}$. I set $y=c$, where $c$ is a constant. Then $\frac{dy}{dx}=0$ and the differential equation becomes $0=\frac{c^3}{x^2}2\frac{c}{x}$ $\frac{c}{x}(\frac{c^2}{x}2)=0$ $c/x=0$ or $c^2/x2=0$ $c=0$ or $c^2=2x$ How to explain in words the equation $c^2=2x$ does not give any constant solution of the differential equation? 
November 13th, 2017, 01:06 AM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,765 Thanks: 623 Math Focus: Yet to find out. 
What do you mean you want to find a constant solution...? Are you trying to find a particular solution?

November 13th, 2017, 07:25 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra 
Duplicate post.

November 13th, 2017, 11:21 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,302 Thanks: 1974 
Differentiation with respect to $x$ requires that $x$ isn't a constant, so c² = 2$x$ isn't possible.

November 20th, 2017, 02:16 AM  #5 
Senior Member Joined: Dec 2015 From: iPhone Posts: 388 Thanks: 61 
Only if $\displaystyle \exists \frac{y_1}{y_2}=C$ Example : equation $\displaystyle y'=(yc_1 )$ has constant solution $\displaystyle y=c_1 $ Last edited by idontknow; November 20th, 2017 at 02:46 AM. 
November 20th, 2017, 09:14 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,302 Thanks: 1974 
What did you mean by $\displaystyle \exists \frac{y_1}{y_2} = C$?


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