Find the particular solution of the differential equation `(dy)/(dx)=(x(2logx+1))/((siny+y cosy))," given that " y=(pi)/(2) " when " x =1.`

Answer

`y sin y=x^(2)log x +(pi)/(2)`

Explanation

`int (siny+y cos y) dy =int(2xlog x +x)dx +C`
`rArr int siny dy +int y cos y dy =int 2x log x dx + int x dx +C`
`rArr (-cosy) + y sin y - int 1*siny dy = 2[(log x) *(x^(2))/(2)-int (1)/(x)*(x^(2))/(2)dx]+(x^(2))/(2)+C`
`rArr y sin y =x^(2)log x +C. " " `...(i)