1st order differential substitution Watch

111davey1
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Hi,
If the substitution is z=y^-2 and want to replace dy/dx by the chain rule:

dy/dx = dy/dz * dz/dx

y=z^-1/2
dy/dz = -1/2z^-3/2

so dy/dx= -1/2z^-3/2 *dz/dx

But... if z=y^-2

and diff each term wrt x:

dz/dx = -2y^-3 *dy/dx

so dy/dx = dz/dx * y^3/-2

Then subbing in for y and rearraning produces same answer..
Is both methods correct?
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RDKGames
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(Original post by 111davey1)
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It's clearer if you just substitute in for y'

y=z^{-1/2} \Leftrightarrow z=y^{-2} so yes both methods are correct and should give the same answer.



If y=z^{-1/2} \Rightarrow y'=-\frac{1}{2}z^{-3/2} z' then subbing in for y' you get

-\frac{1}{2}z^{-3/2} z' = \frac{dy}{dz} \cdot z'

\Rightarrow -\frac{1}{2}z^{-3/2} = \frac{dy}{dz}





If z=y^{-2}\Rightarrow z'=-2y^{-3}y' \Rightarrow y'=-\frac{1}{2}y^3 z' = \frac{dy}{dz} \cdot z'

\Rightarrow -\frac{1}{2}y^3 = \frac{dy}{dz}




Now the only way where \frac{dy}{dz}=-\frac{1}{2}y^3 = -\frac{1}{2}z^{-3/2} is if y=z^{-1/2}
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