Question

What is the general solution of the differential equation `xyy'=x^2+1` ?

Answer 1

`y^2 = x^2 + 2ln|x| + A`

Explanation:

The differential equation

`xyy'=x^2+1`

is a First Order linear separable Differential Equation which can be solved simply by rearranging and collection term in `x` on the RHS and term in `y` on the LHS;

`ydy/dx = (x^2+1)/x`

And now we "separate the variables" to get;

`int \ y \ dy = int \ x+1/x \ dx`

Which is trivial to integrate to get:

`\ \ 1/2y^2 = 1/2x^2 + ln|x| + C`

`:. y^2 = x^2 + 2ln|x| + 2C`
`:. y^2 = x^2 + 2ln|x| + A`