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Question

Solve the differential equation (1+x2)dydx+y=etan1x.

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Solution

(1+x2)dydx+y=etan1x
dydx+y1+x2=etan1x1+x2
It is linear differential equation of first order.
Comparing with standard linear differential equation.
dydx+P(x)y=Q(x)
P(x)=11+x2;Q(x)=etan1x1+x2
Integrating factor(IF) = ePdx=e11+x2dx=etan1x
Solution of LDE,
y.IF=IFQ(x)dx+C
y.etan1x=etan1x.etan1x1+x2dx+C
y.etan1x=(etan1x)21+x2dx+C---(1)
To solving (etan1x)21+x2dx
Put etan1x=t
or etan1x.11+x2dx=dt
Therefore, etan1x.etan1x1+x2dx=tdt
=t22+C(etan1x)22+C
Substituting in (1),
y.etan1x=(etan1x)22+C.

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