Question

Solve the given differential equation by using an appropriate substitution. The DE is a Bernoulli equation.

x dy/dx + y = 1/y^2

Answer 1

Multiplying both sides by
`y^2` gives

`xy^2(\mathrm dy)/(\mathrm dx)+y^3=1`

so that substituting
`v=y^3` and hence
`(\mathrm dv)/(\mathrm dv)=3y^2(\mathrm dy)/(\mathrm dx)` gives the linear ODE,

`\frac x3(\mathrm dv)/(\mathrm dx)+v=1`

Now multiply both sides by
`3x^2` to get

`x^3(\mathrm dv)/(\mathrm dx)+3x^2v=3x^2`

so that the left side condenses into the derivative of a product.

`(\mathrm d)/(\mathrm dx)[x^3v]=3x^2`

Integrate both sides, then solve for
`v`, then for
`y`:

`x^3v=\displaystyle\int3x^2\,\mathrm dx`

`x^3v=x^3+C`

`v=1+\frac C{x^3}`

`y^3=1+\frac C{x^3}`

`\boxed{y=\sqrt[3]{1+\frac C{x^3}}}`