Question

Don't have answer sheet & like confirmation on my calc:

x dy/dx sin y/x = y sin y/x – x
let y=vx then v = y/x & dy/dx = v + (x)dv/dx

x[v + (x)dv/dx] sin(v) = vxsin(v) – x substitute values
(vx + (x²)dv/dx) = vx – [x/sin(v)] Divided by sin(v)
(x²)dv/dx = - x/sin(v) vx cancel on both sides
(1) dv/dx = - 1/[(x)sin(v)] divide by (x²) on both sides
sin(v) dv/dx = - 1/x multiply by sin(v) on both sides
∫sin(v) dv = - ∫1/x dx
- cos(v) = - lnǀxǀ + C
- cos(y/x) = - lnǀxǀ + C
cos(y/x) - lnǀxǀ + C = 0

Answer 1

youve solved the DE correctly
i got the same result with the same basic method
however i found it easier to rearrange the equation before substitution

x dy/dx sin y/x = y sin y/x – x

dy/dx sin y/x = y/x sin y/x -1

dy/dx = y/x - 1/sin y/x

dy/dx = y/x- csc y/x
now subsituting y/x=v
v + xdv/dx = v-csc v

xdv/dx = - csc v

sin v dv = -dx/x