Question

For each of the differential equation in given question find the general solution.​​​​
$se{c}^{2}xtanydx+se{c}^{2}ytanxdy=0$

Answer 1

Given, $se{c}^{2}xtanydx+se{c}^{2}ytanxdy=0$
On separating the variables, we get
$\Rightarrow se{c}^{2}xtanydx=-se{c}^{2}ytanxdy\Rightarrow \frac{se{c}^{2}x}{}tanxdx=-\frac{se{c}^{2}y}{}tanydy$
On integarting both sides, we get $\int \frac{se{c}^{2}x}{}tanxdx=-\int \frac{se{c}^{2}y}{}tanydyLettanx=u\Rightarrow se{c}^{2}x=\frac{du}{dx}\Rightarrow dx=\frac{du}{se{c}^{2}x}andtany=v\Rightarrow se{c}^{2}y=\frac{dv}{dy}\Rightarrow dy=\frac{dv}{se{c}^{2}y}\int \frac{se{c}^{2}x}{u}\frac{du}{se{c}^{2}x}=-\int \frac{se{c}^{2}y}{v}\frac{dv}{se{c}^{2}y}\Rightarrow \int \frac{du}{u}=-\int \frac{dv}{v}\Rightarrow log|u|=-log|v|+log|C|\Rightarrow log|tanx|=-log|tany|+log|C|\Rightarrow log|tanxtany|=log|C|[\because logm+logn=logmn]\Rightarrow tanx.tany=C$
which is the required general solution.