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Question

Solve the differential equation (tan1 xy) dx=(1+x2)dy.


Solution

(tan1xy)dx=(1+x2)dy

dydx=tan1xy1+x2

dydx+y1+x2=tan1x1+x2      ...(i)

I.F. = e11+x2dx=etan1x

Multiplying both sides of (i) by etan1x , we have

dydx.etan1x+y1+x2.etan1x=tan1x1+x2.etan1x

Integrating both sides w.r.t. x, we have y.etan1x=tan1x1+x2.etan1xdx+c

Put tan1x=t11+x2dx=dt

y.et=ett dt+C 

y.et=et(t1)+C
y.etan1x=etan1x(tan1x1)+C
y=tan1x1+C.etan1x
 

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