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jeka57 [31]
3 years ago
12

Determine an appropriate solution to solve xdy/dx=yln(xy)

Mathematics
1 answer:
zloy xaker [14]3 years ago
7 0
x\dfrac{\mathrm dy}{\mathrm dx}=y\ln(xy)

Let v=xy, so that \dfrac{\mathrm dv}{\mathrm dx}=y+x\dfrac{\mathrm dy}{\mathrm dx}, or \dfrac{\mathrm dv}{\mathrm dx}-\dfrac vx=x\dfrac{\mathrm dy}{\mathrm dx}. Then you have

\dfrac{\mathrm dv}{\mathrm dx}-\dfrac vx=\dfrac vx\ln v
\dfrac{\mathrm dv}{v(\ln v+1)}=\dfrac{\mathrm dx}x

Integrate both sides to get

\ln|\ln v+1|=\ln|x|+C
\ln v+1=Cx
v=e^{Cx-1}

then back-substitute to find the solution for y.

xy=e^{Cx-1}\implies y=\dfrac{e^{Cx-1}}x
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