To solve the given differential equation
(1+y2)‌dx=(tan−1y−x)dy, we rearrange it into a form that allows us to separate variables or use another method to simplify.
Dividing both sides by
dy and rearranging the terms, we get:
(1+y2)‌+x=tan−1yNow, let's try using an integrating factor method. Rewriting the equation:
‌+‌=‌We observe that the integrating factor,
µ(y), for this differential equation must satisfy:
‌=‌Solving this, we find:
µ(y)=e∫‌dy=etan−1y Multiplying through the differential equation by
etan−1y :
etan−1y‌+etan−1y‌=etan−1y‌This simplifies to:
‌(etan−1yx)=etan−1y‌Since the derivative of
tan−1y with respect to
y is
‌, replacing in the equation gives:
‌(etan−1yx)=etan−1y(tan−1y)(‌)This simplifies further:
‌(etan−1yx)=etan−1yd(tan−1y)Integrating both sides with respect to
y gives:
etan−1yx=∫d(etan−1y)+C=etan−1y+C Finally, solving for
x :
x=1+Ce−tan−1yTo match this result with the options given, we notice if we let
C−1 be represented as a new constant, say
c′, then:
x=tan−1y−1+ce−tan−1yThus, the correct option based on the above development should be:
Option B:
x=tan−1y−1+ce−tan−1y