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Elena-2011 [213]
3 years ago
12

A Bernoulli differential equation is one of the form dxdy+P(x)y=Q(x)yn Observe that, if n=0 or 1, the Bernoulli equation is line

ar. For other values of n, the substitution u=y1−n transforms the Bernoulli equation into the linear equation dxdu+(1−n)P(x)u=(1−n)Q(x) Use an appropriate substitution to solve the equation xy+y=4xy2
Mathematics
1 answer:
aleksandr82 [10.1K]3 years ago
5 0

It's unclear whether the ODE is

xy'+y=4xy^2

or

xy+y'=4xy^2

If the first case, then divide through both sides by y^2 to get

xy^{-2}y'+y^{-1}=4x

then substitute z=y^{-1}, so that z'=-y^{-2}y'. Then

-xz'+z=4x\implies-\dfrac1xz'+\dfrac1{x^2}z=\left(-\dfrac1xz\right)'=\dfrac4x

\implies-\dfrac1xz=\displaystyle\int\frac4x\,\mathrm dx

\implies z=-4x\ln|x|+Cx

\implies\boxed{y=\dfrac1{Cx-4x\ln|x|}}

If the second, then dividing through by y^2 gives

xy^{-1}+y^{-2}y'=4x

The same substitution as before gives

xz-z'=4x\implies e^{-x^2/2}z'-xe^{-x^2/2}z=\left(e^{-x^2/2}z\right)'=-4xe^{-x^2/2}

\implies e^{-x^2/2}z=\displaystyle-4\int xe^{-x^2/2}\,\mathrm dx

\implies z=4+Ce^{x^2/2}

\implies\boxed{y=\dfrac1{4+Ce^{x^2/2}}}

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