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SCALC8 9.5.024. My Notes A Bernoulli differential equation (named after James Bernoulli) is of the form dy dx + P(x)y = Q(x)yn. Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y1 − n transforms the Bernoulli equation into the linear equation du dx + (1 − n)P(x)u = (1 − n)Q(x). Use the substitution u = y1 − n to solve the differential equation. xy' + y = −5xy2

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[tex]xy'+y=-5xy^2[/tex]

Divide through the ODE by the largest power of [tex]y[/tex], assuming [tex]y\neq0[/tex]:

[tex]xy^{-2}y'+y^{-1}=-5x[/tex]

By the chain rule, [tex](y^{-1})'=-y^{-2}y'[/tex]. So substitute [tex]z=y^{-1}[/tex] and [tex]-z'=y^{-2}y'[/tex] to get

[tex]-xz'+z=-5x[/tex]

which is linear in [tex]z[/tex]. Multiply both sides by [tex]-\frac1{x^2}[/tex]:

[tex]\dfrac{z'}x-\dfrac z{x^2}=\dfrac5x[/tex]

Now the left side is the derivative of a product, so we can condense this as

[tex]\left(\dfrac zx\right)'=\dfrac5x[/tex]

Integrate both sides with respect to [tex]x[/tex]:

[tex]\dfrac zx=5\ln|x|+C[/tex]

[tex]z=5x\ln|x|+Cx[/tex]

Solve for [tex]y[/tex]:

[tex]\dfrac1y=5x\ln|x|+Cx\implies\boxed{y=\dfrac1{5x\ln|x|+Cx}}[/tex]

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