Question

Find the particular solution of the differential equation : (x-siny)dy+(tany)dx=0 given that y=0 when x=0

Answer 1

Solution:

(x-sin y)dy +(tan y)dx=0

$x-sin y +tan y\frac{dx}{dy}=0\\\\ \frac{dx}{dy} +\frac{x}{tany}=\frac{siny}{tan y}\\\\\frac{dx}{dy} +xcoty}=cosy\\\\ {\text{integrating factor}}=e^{\int\limits {coty} \, dx =e^{log(siny)}=sin y$

Multiplying both sides by sin y

$sin y(\frac{dx}{dy} +xcoty})=siny\times cosy$

integrating both sides with respect to y

$xsiny=\int\limits {\frac{sin2y}{2}} \, dx \\\\ x\times siny=\frac{-cos2y}{4}+C, as,{\text{given that y=0 when x=0}},C=\frac{1}{4}\\\\  x\times siny=\frac{-cos2y}{4}+\frac{1}{4}$

is the particular solution of the differential equation :

As, sin2y=2 sin y cos y, cos 0°=1