Can someone solve this? :)

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1 Answer
May 13, 2018

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