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November 12th, 2017, 10:59 PM   #1
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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 $c^2/x-2=0$
$c=0$ or $c^2=2x$

How to explain in words the equation $c^2=2x$ does not give any constant solution of the differential equation?
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November 13th, 2017, 12:06 AM   #2
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What do you mean you want to find a constant solution...? Are you trying to find a particular solution?
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November 13th, 2017, 06:25 AM   #3
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Duplicate post.
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November 13th, 2017, 10:21 AM   #4
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Differentiation with respect to $x$ requires that $x$ isn't a constant, so c² = 2$x$ isn't possible.
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November 20th, 2017, 01:16 AM   #5
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Only if $\displaystyle \exists \frac{y_1}{y_2}=C$
Example : equation $\displaystyle y'=(y-c_1 )$ has constant solution $\displaystyle y=c_1 $

Last edited by idontknow; November 20th, 2017 at 01:46 AM.
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November 20th, 2017, 08:14 AM   #6
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What did you mean by $\displaystyle \exists \frac{y_1}{y_2} = C$?
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