Question

write the ordinary differential equation (1+sin y)dx = (2ycosy-x(secy-tan y))dy

Answer 1

$\left(\mathrm{1}+\mathrm{sin(}y\mathrm{)}\right)dx{}{}\ \ =\ \ \left(\mathrm{2}y\ \mathrm{cos}\left(y\right)\ \ -\ \ x\left(\mathrm{sec}\left(y\right)\ \ -\ \ \mathrm{tan}\left(y\right)\right)\right)dy \\ \\ \frac{dx}{dy}\ =\ \ \frac{\mathrm{2}y\ \mathrm{cos}\left(y\right)\ \ -\ \ x\left(\mathrm{sec}\left(y\right)\ \ -\ \ \mathrm{tan}\left(y\right)\right)}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)} \\ \\ \frac{dx}{dy}\ +\frac{\ x\left(\mathrm{sec}\left(y\right)\ \ -\ \ \mathrm{tan}\left(y\right)\right)}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}=\ \ \frac{\mathrm{2}y\ \mathrm{cos}\left(y\right)\ \ }{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}\ \ \ \ \ \left(First\ \ Order\ \ Linear\ \ Equation\right) \\ \\ IF\ \ =\ \ e^{\int{\frac{\ \left(\mathrm{sec}\left(y\right)\ \ -\ \ \mathrm{tan}\left(y\right)\right)}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}dy}}\ \ \ =\ \ e^{\frac{-\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}} \\ \\ x\ \ e^{\frac{-\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \ \ =\ \ \int{\frac{\mathrm{2}y\ \mathrm{cos}\left(y\right)\ \ }{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}e^{\frac{-\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \ dy\ \ \ \ +\ \ C \\ \\ x\ \ =\ \ \ \ e^{\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \int{\frac{\mathrm{2}y\ \mathrm{cos}\left(y\right)\ \ }{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}e^{\frac{-\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \ dy\ \ \ \ +\ \ \ Ce^{\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \\ \\ x\left(y\right)=\ \ \ \ e^{\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \int{\frac{\mathrm{2}y\ \mathrm{cos}\left(y\right)\ \ }{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}e^{\frac{-\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}\ \ dy\ \ \ \ +\ \ \ Ce^{\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{sin}\left(y\right)\right)}}$