Solve Differential Equation by means of Separating Variables

Examples

Ex1 ${dy \over dx}={x^{2} \over y^{2}}$

$y^{2}*dy=x^{2}*dx$

${\int y^{2}*dy}={\int x^{2}*dx}$

${\int y^{2}*dy}={y^{3} \over 3}$

${\int x^{2}*dx}={x^{3} \over 3}$

But one side of the equation needs to add a constant c.

${y^{3} \over 3}={x^{3} \over 3}+c$

$y^{3}=x^{3}+3c$

constant times 3 will still be constant so 3c-> c.

${\sqrt[{3}]{y^{3}}}.={\sqrt[{3}]{x^{3}+c}}$

Ex2 y' = xy ${dy \over dx}=xy$

$dy=xy*dx$

${dy \over y}=x*dx$

$\int {1 \over y}dy=\int x*dx$

$\int {1 \over y}dy=ln|y|$

$\int x*dx={x^{2} \over 2}+c$

reference

https://www.youtube.com/watch?v=C7nuJcJriWM&list=PLEjLk3Wl8akWPgisw-u9jrmdN67dgPibe&index=33