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(x(dy)/(dx) -y ) sin (y /x) =x^(2)e^(x) ...

`(x(dy)/(dx) -y ) sin (y /x) =x^(2)e^(x) , y = vx `

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Solution of the differential equation (x(dy)/(dx)-y)sin((y)/(x))=x^(2)e^(x) is

(dy) / (dx) = (sin x) / (e ^ (y))

(dy) / (dx) = (y) / (x) + sin ((y) / (x))

(dy)/(dx) = e^(2x-y) + x^(3) e^(-y)

(dy)/(dx)=e^(-y)*sin x+e^((x-y))

(dy)/(dx)=e^(-y)sin x+e^((x-y))

Solve (dy)/(dx)= e^(x-y) + x^2 e^(-y) .

using the subsitiution y = vx : (1) (x(dy)/(dx) -y)^(e^(y/x) =x^(2) cos x

Find (dy)/(dx) if y =e^(x) sin 2x