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"`

∴ (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.