Show that

a) Y= x exp-x statisfies the equation xy'=(1-x)y

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- Feb 10th 2009, 09:38 PMhomerbDerivatives Of inverse functions; Derivatives and integrals involving exponential fun
Show that

a) Y= x exp-x statisfies the equation xy'=(1-x)y - Feb 10th 2009, 09:42 PMo_O
Use product rule to get: $\displaystyle y = xe^{-x} \ \ \Rightarrow \ \ y' = e^{-x} - xe^{-x}$

So: $\displaystyle \text{LHS} = xy' = x \left(e^{-x} - xe^{-x}\right) = \cdots$

And: $\displaystyle \text{RHS} = (1-x)y = (1-x)(xe^{-x}) = \cdots$

Are they equal? If so, you're done. - Feb 10th 2009, 09:50 PMhomerb
do I have to use ln?

or what is the next step - Feb 10th 2009, 09:53 PMo_O
What do you mean? We just want to show $\displaystyle xy' = (1-x)y$.

We've got an expression for $\displaystyle xy'$ and one for $\displaystyle (1-x)y$. Show that they're equal by simplifying and you're done. - Feb 10th 2009, 09:54 PMhomerb
oh I see thanks, I thinking something else?