Question

Can someone solve this? :)

Answer 1

This is a first order linear differential equation

Explanation:

Start off by rearranging the equation. Get the `dx/dy` or `dy/dx` on one side and the other terms on one side.

In this question we rearrange the equation by getting `dx/dy` on one side. After splitting the numerator and solving , we get ,

`dx/dy` + `(xy)/(1+y^2)` = `siny/sqrt(1+y^2)`

-> This is a linear differential equation of the type
`dx/dy` + `Px` = `Q`
(Here P and Q are functions of y)
In this equation `P` = `y/(1+y^2)`

The integrating factor for the equation is :
-> `e^(int(Pdx))`
->`e^(int((ydy)/(1+y^2)))`
->Solving the integral using substitution
-> Let `1+y^2` = t
->Upon differentiating , you get , `ydy=dt/2`
You end up with `e^(1/2(log(1+y^2))`
Integrating factor is ->`sqrt(1+y^2)`**

Multiplying this factor on both sides and then integrating ,

-> `x.sqrt(1+y^2)` = `int(siny/sqrt(1+y^2)).sqrt(1+y^2)`
Cancelling `sqrt(1+y^2)` you get ,

`x.sqrt(1+y^2)` = `int(sinydy)`
Upon integrating ,
`x.sqrt(1+y^2)` = `-cosy + C`

Therefore the final answer is :
-> `x.sqrt(1+y^2)` + `cosy` = `C`