Dy/dx = y/x , y(1) = −2

Question

asked Mar 7, 2020 184k views

1 Answer

Rokia · Answer 1 · 2020-03-12T03:39:47+0000

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