Answer :

Given,


(tan-1 y – x)dy = (1 + y2)dx



Clearly, this is a linear differential equation. Comparing with the standard form



A solution of such equation is given by:


x(I.F) = where I.F = integrating factor


We get P(y) = & Q(y) =


Integrating factor I.F is given :


I.F =


We know that:


I.F =


The solution is given as:



…(1)


Where I =


Let tan-1y = u


du =


I =


Using integration by parts:


I =


I =


I =


the solution is given using equation 1:




OR


Given,


…(1)


Clearly, the equation is homogeneous(can be observed directly)


We know that for solving a homogeneous we differential equation,


We take y = vx.


As y = vx


Differentiating both sides w.r.t x we get




equation 1 can be rewritten as-






Integrating both sides we get-








The above equation gives the general solution. For a particular solution, we need to find the value of C.


As given that at x = 0 ,y = 1



C = 0.


Particular solution at x = 0 and x =1 is given by:



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