Question

The solution of the differential equation `dy/dx = tan(y/x) + (y/x)` is ______.

पर्याय

• `cos(y/x) = cx`

• `sin (y/x) = cx`

• `cos(y/x) = cy`

• `sin(y/x) = cy`

Answer 1

The solution of the differential equation `dy/dx = tan(y/x) + (y/x)` is `underlinebb(sin (y/x) = cx)`.

Explanation:

We have, `dy/dx = tan(y/x) + (y/x)`  ...(i)

Since, it is a homogeneous differential equation

After putting, y = Vx, we get

`dy/dx = V + x (dV)/dx`

`V + x(dV)/dx` = tan V + V  ...[From (i)]

`\implies x (dV)/dx` = tan V

`\implies 1/tanV dV = 1/x dx`

After integrating on both sides, we get

`int 1/tanV dV = int 1/x dx`

`\implies int cot V dV` = log x + log c

`\implies` log sin V = log(xc)

`\implies` sin v = xc

Hence, `sin(y/x)` = xc