Education • All: 2.1: Separable Variables

SEPARABLE VARIABLES

Separate dy and dx by having dy and y variables in left hand side of the equal sign and dx and x variables on the right hand side of the equal sign.

Example (1):

$\large \\dy-(y-1)^{2}dx=0\;\;(1.Given\;equation) \\\\dy=(y-1)^{2}dx\;\;\;\;(2.Separate\;variables\;on\;each\;side) \\\\\int \frac{dy}{(y-1)^{2}}=\int dx\;\;\;\;(3.Integrate \;both \;sides) \\\\\frac{-1}{y-1}=x+C\;\;\;\;(5.Solve\;for\;y) \\\\1-\frac{1}{x+C}=y\;\;\;\;(4.Explicit \;solution)$

Example (2): With integration by parts)

$\dpi{120} \\e^{x}y\frac{dy}{dx}=e^{-y}+e^{-2x-y}\;\;\;\;(1.Given\;equation) \\\\e^{x}y\frac{dy}{dx}=e^{-y}(1+e^{-2x})\;\;\;\;(2.Simplify\;variables) \\\\ye^{y}dy=\frac{1+e^{-2x}}{e^{x}}dx\;\;\;\;\;\;(3.Separate\;variables\;on\;each\;side) \\\\\int ye^{y}dy=\int(e^{-x}+e^{-3x}) dx\;\;\;\;(4.Integrate \;both \;sides\;by\;parts) \\\\ye^{y}-e^{y}=-e^{-x}-\frac{1}{3}e^{-3x}+C\;\;\;\;(5.Solve\;for\;C) \\\\ye^{y}-e^{y}+\frac{1}{e^{x}}+\frac{1}{3e^{x}}=C\;\;\;\;(4.Implicit \;solution)$

Example (3): With integration by substitution)

$\\\\\frac{dy}{dx}=(\frac{2y+3}{4x+5})^{2}\;\;\;\;(1.Given\;equation) \\\\\frac{dy}{(2y+3)^{2}}=\frac{dx}{(4x+5)^{2}}\;\;\;\;\;\;(2.Separate\;variables\;on\;each\;side) \\\\\int \frac{dy}{(2y+3)^{2}}=\int\frac{dx}{(4x+5)^{2}}\;\;\;\;(4.Integrate \;both \;sides\;by\;substitution) \\\\\frac{-1}{2(2y+3)}=\frac{-1}{4(4x+5)}+C\;\;\;\;(5.Solve\;for\;C) \\\\\frac{-1}{2y+3}=\frac{-1}{8x+10}+C\;\;\;\;(4.Implicit \;solution)$

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2.1: Separable Variables

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