Differential Equation Transformation help

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Zacken
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#1
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Using the substitution y-x = u, the differential equation

\displaystyle \frac{\mathrm{d} y}{\mathrm{d} x} + e^{y-x} = 1

is transformed to

\displaystyle \frac{\mathrm{d} u}{\mathrm{d} x} + e^u = 0

But I can't seem to find a general solution for the transformed equation.

Using an integrating factor of I = e^x yields

\displaystyle e^x e^u = \int 0 \, \mathrm{d}x

\Rightarrow \displaystyle e^{x+u} = a

\Rightarrow \displaystyle u = -x + \ln a

But the required solution is u = -\ln(x + a)
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Kevin De Bruyne
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#2
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(Original post by Zacken)
Using the substitution y-x = u, the differential equation

\displaystyle \frac{\mathrm{d} y}{\mathrm{d} x} + e^{y-x} = 1

is transformed to

\displaystyle \frac{\mathrm{d} u}{\mathrm{d} x} + e^u = 0

But I can't seem to find a general solution for the transformed equation.

Using an integrating factor of I = e^x yields

\displaystyle e^x e^u = \int 0 \, \mathrm{d}x

\Rightarrow \displaystyle e^{x+u} = a

\Rightarrow \displaystyle u = -x + \ln a

But the required solution is u = -\ln(x + a)
I would suggest separating the variables instead of using an integrating factor.
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Zacken
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(Original post by SeanFM)
I would suggest separating the variables instead of using an integrating factor.
...I hate myself, that was so obvious. Thank you so much!

On the flip side though, why didn't the integrating factor method work?
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Kevin De Bruyne
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(Original post by Zacken)
...I hate myself, that was so obvious. Thank you so much!

On the flip side though, why didn't the integrating factor method work?
I'm not sure what would differentiate to give du/dx + e^u, but it's not e^x * e^u. You'd get a few more exponent terms when you use the product rule. But apart from that I don't really know.
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Zacken
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(Original post by SeanFM)
I'm not sure what would differentiate to give du/dx + e^u, but it's not e^x * e^u. You'd get a few more exponent terms when you use the product rule. But apart from that I don't really know.
I'd suspect it was function continuity but Q(x) = 0 is continuous and I'm not sure what assumption I'm violating for this differential equation to not be applicable to the integrating factor theorem.

Oh well, thanks anyway! +Rep.
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