Question

Dy/dx = y/x , y(1) = −2

Answer 1

Start with

`(dy)/(dx)=(y)/(x)`

Separate the variables:

`(dy)/(y) = (dx)/(x)`

Integrate both parts:

`\displaystyle \int (dy)/(y) = \int(dx)/(x)`

Which implies

`\log(y) = \log(x)+c`

Solving for y:

`y = e^(\log(x)+c) = e^(\log(x))e^c=xe^c`

Since
`e^c` is itself a constant, let's rename it
`c_1`.

Fix the additive constant imposing the condition:

`y(1) = c_1\cdot 1 = -2\iff c_1=-2`

So, the solution is

`y(x) = -2x`