Question

Solve the differential equation `2xlnx dy/dx + y = 0`?

Answer 1

`y = A/sqrt(lnx)`

Explanation:

We have:

`2xlnx dy/dx + y = 0` ..... [A]

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

`dy/dx + P(x)y=Q(x)`

So rewrite the equations in standard form as:

`dy/dx + 1/(2xlnx)y = 0 ..... [B]`

Then the integrating factor is given by;

`I = e^(int P(x) dx)`
`\ \ = exp(int \ 1/(2xlnx) \ dx)`
`\ \ = exp( 1/2ln|lnx| )` (see notes at end)
`\ \ = exp( lnsqrt(|lnx|))`
`\ \ = sqrt(lnx))`

And if we multiply the DE [B] by this Integrating Factor, `I`, we will have a perfect product differential form of `[A]`;

`sqrt(lnx)dy/dx + sqrt(lnx)1/(2xlnx)y = 0`
`:. d/dx (ysqrt(lnx)) = 0`

`:. ysqrt(lnx)) = A`

Which we can rearrange to get:

`:. y = A/sqrt(lnx)`

Which, is the General Solution.