Question

Solve the following differential equation:

`x * dy/dx - y + x * sin(y/x) = 0`

Answer 1

`"x" "dy"/"dx" - "y" + "x sin"("y"/"x") = 0`   ...(1)

Put y = vx

∴ `"dy"/"dx" = "v + x" "dv"/"dx" and "y"/"x" = "v"`

∴ equation (1) becomes,

`x("v + x""dv"/"dx") - "vx + x sin v" = 0`

∴ `"vx" + "x"^2 "dv"/"dx" - "vx" + "x sin v" = 0`

∴ `"x"^2 "dv"/"dx" + "x sin v" = 0`

∴ `1/"sin v" "dv" + 1/"x" "dx" = 0`

Integrating, we get

∴ `int "cosec v dv" + int1/"x" "dx" = "c"_1`

∴ `log |"cosec v - cot v"| + log |"x"| = log "c"`, where c₁ = log c

∴ `log |"x" ("cosec v" - "cot v")| = log "c"`

∴ `"x"(1/(sin"v") - (cos "v")/(sin"v")) = "c"`

∴ x(1 - cos v) = c sin v

∴ `"x"[1 - cos("y"/"x")] = "c sin"("y"/"x")`

This is the general solution.

Answer 2

`x * dy/dx - y + x * sin(y/x) = 0`

Put y = vx

⇒ `dy/dx = v + x (dv)/dx`

∴ The given equation becomes, 

`x (v + x (dv)/dx) - vx + x sin v = 0`

∴ `vx + x^2 (dv)/dx - vx + x sin v = 0`

∴ `x^2 (dv)/dx = -x sin v`

∴ `(dv)/(sin v) = - dx/x`    ...(variable separable form)

Integrating on both sides, we get,

`int cosec  v  dv = -int 1/x dx`

∴ log (cosec v − cot v) = − log x + log c

∴ log (cosec v − cot v) = `log (c/x)`

∴ cosec v − cot v = `c/x`

∴ `x [cosec (y/x) - cot (y/x)] = c` is the required solution.