Q25 of 45 Page 1

Find the particular solution of the differential equation

(tan–1 y – x) dy = (1 + y2) dx, given that when x = 0, y = 0.

Given,


(tan-1 y – x)dy = (1 + y2)dx


⇒


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



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 =


∴ Solution is given as:



⇒ …(1)


Where I =


Let tan-1y = u


⇒ du =


∴ I =


Using integration by parts:


I =


∴ I =


⇒ I =


∴ solution is given using equation 1:



⇒ …(2)


As we have to find particular solution, so we need to find value of C.


Given: x = 0, y = 0


∴ putting the values in eq (2)


0 = tan-1(0) – 1 +


⇒ C = 1


∴ particular solution is –



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