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#