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 ${dy \over dx}=xy$

$dy=xy*dx$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {dy \over y} = x * dx}}

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

reference

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