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Question

Solve : (1+y2)dx=(tan1yx)dy.

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Solution

Given that(1+y2)dx=(tan1yx)dy dxdy=tan1y1+y2x1+y2 dxdy+x1+y2=tan1y1+y2The differential equation is in the form dxdy+P(y)x=Q(y) where P(y)=11+y2 and Q(y)=tan1y1+y2So, Integration factor, (I.F.)=e11+y2dy=etan1ysolution is given as : x×etan1y=etan1y×tan1y1+y2dy+CPut tan1y=t11+y2dy=dt. xetan1y=ettdt+Cxetan1y=tetdt(ddt(t)etdt)+C xetan1y=tetet+Cxetan1y=etan1y(tan1y1)+C or, x=tan1y1+Cetan1y is the required solution

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