Question

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

Answer 1

`xy'+y=-5xy^2`

Divide through the ODE by the largest power of
`y`, assuming
`y\\eq0`:

`xy^(-2)y'+y^(-1)=-5x`

By the chain rule,
`(y^(-1))'=-y^(-2)y'`. So substitute
`z=y^(-1)` and
`-z'=y^(-2)y'` to get

`-xz'+z=-5x`

which is linear in
`z`. Multiply both sides by
`-\frac1{x^2}`:

`\frac{z'}x-\frac z{x^2}=\frac5x`

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

`\left(\frac zx\right)'=\frac5x`

Integrate both sides with respect to
`x`:

`\frac zx=5\ln|x|+C`

`z=5x\ln|x|+Cx`

Solve for
`y`:

`\frac1y=5x\ln|x|+Cx\implies\boxed{y=\frac15x\ln}`