Examples of differential equations: Difference between revisions

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== Separable first-order ordinary differential equations ==

{{see also|Separable partial differential equation}}

Equations in the form <{{math>\frac |1=''{dy}{sfrac|dy|dx} }''= ''f''&hairsp;(''x'')''g''(''y'')</math>}} are called separable and solved by <{{math>\frac |1={{sfrac|''dy}{''|''g''(''y'')}} = ''f''&hairsp;(''x'')\, ''dx</math>''}} and thus

:<math>\int\frac {dy}{g(y)} = \int f(x) \, dx</math>. Prior to dividing by <math>g(y)</math>, one needs to check if there are stationary (also called equilibrium)

Prior to dividing by {{math|''g''(''y'')}}, one needs to check if there are stationary (also called equilibrium) solutions with constant {{mvar|y}} satisfying {{math|1=''g''(''y'') = 0}}.

solutions <math>y=\text{constant}</math> satisfying <math>g(y)=0</math>.

==Separable (homogeneous) first-order linear ordinary differential equations==