**Ordinary** - involves only *one* independent variable $ \frac{dy}{dt}+2ty=0 $ (see [[Partial Differential Equations (PDEs)]] for more than one independent variable) ## First Order ODEs **First Order** - derivatives of degree one #### Solution Methods - [[Method of Separation]] - [[Method of Integrating Factors]] ## Higher Order ODEs ### Homogeneous A combination of $y$ and its derivatives that sum exactly to zero: $ \alpha_n\frac{d^ny}{dt^n}+\alpha_{n-1}\frac{d^{n-1}y}{dt^{n-1}}+\dots+\alpha_1\frac{dy}{dt}+\alpha_0y=0 $ #### Solution Methods - Finding roots of the [[Characteristic Equation|characteristic equation]] ### Non-homogeneous This type of equation is generally organized as: $ ay''+by'+cy=f(t) $ #### Solution Methods [[Method of Undetermined Coefficients]]