1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# Solve the differential equation:(tan−1y−x)dy=(1+y2)dx.

Open in App
Solution

## On rearranging the equation becomes:(tan−1y−x)dy=(1+y2)dx⇒dxdy+x1+y2=tan−1y1+y2.This is in the form dxdy+P(y).x=Q(y), so it is a non homogeneous linear differential equation of the first order.the integrating factor is: I.F.=e∫dy1+y2=etan−1yMultiplying both sides of the equation with integrating factor, we get:etan−1y(dxdy+x1+y2)=etan−1ytan−1y1+y2⇒etan−1ydxdy+etan−1yx1+y2=etan−1ytan−1y1+y2⇒ddy(xetan−1y)=etan−1ytan−1y1+y2Integrating by parts, by taking tan−1y as first function and etan−1y1+y2 as second function we get:⇒xetan−1y=∫etan−1ytan−1y1+y2dy=tan−1y∫etan−1y1+y2dy−∫d(tan−1y)(∫etan−1y1+y2dy)=tan−1y.etan−1y−∫dy1+y2(∫etan−1y1+y2dy)⇒xetan−1y=tan−1y.etan−1y−etan−1y+C⇒x=tan−1y−1+Ce−tan−1y

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
General and Particular Solutions of a DE
MATHEMATICS
Watch in App
Join BYJU'S Learning Program