Question

What is a particular solution to the differential equation `dy/dx=(4sqrtylnx)/x` with `y(e)=1`?

Answer 1

`y = ln^4x`

Explanation:

We have:

`dy/dx = (4sqrt(y)lnx)/x`

Which is a first order linear separable Differential Equation, so we can rearrange to get:

`1/sqrt(y) dy/dx = (4lnx)/x`

and separate the variables to get:

`int \ y^(-1/2) \ dy = int \ (4lnx)/x \ dx`

And then we can integrate to get:

`y^(1/2)/(1/2) = (4)(ln^2x/2) + C`
`:. 2sqrt(y) = 2ln^2x + C`

Using `y(e)=1` we get:

`:. 2 = 2ln^2e + C`
`:. C=0`

Hence the particular solution is:

`\ \ \ 2sqrt(y) = 2ln^2x`
`:. sqrt(y) = ln^2x`
`:. \ \ \ y = ln^4x`

Validation:
1. `x=e => y = ln^4e = 1 \` QED
2. `dy/dx = (4ln^3x)/x = (4lnx)/x * ln^2x = (4sqrt(y)lnx)/x \` QED