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Differential equation (DE) 
An equation containing derivatives of one or more unknown functions (or dependent variables) with respect to one or more independent variables
Examples of DE 
𝑑𝑦 𝑑𝑥 =𝑐𝑜𝑠𝑥
𝑑2𝑦 𝑑𝑥2 +��2𝑦 =0
(𝑥2 +𝑦2)𝑑𝑥 −2𝑥��𝑑𝑦 = 0
𝜕2𝑉 𝜕𝑥2
(𝑑2𝑤 𝑑𝑥2 +𝜕2𝑉 𝜕��2 3 ) =0 −𝑥𝑦𝑑𝑤 𝑑𝑥
𝑥𝜕𝑓 𝜕𝑥 +𝑦𝜕�� 𝜕𝑦 +𝑤=0
Ordinary Differential Equation (ODE) 
Contains only ordinary derivatives of one or more functions with respect to a single independent variable
Partial Differential Equation (PDE)
Involves partial derivatives of one or more unknown functions of two or more independent variables
Order of DE 
The order of the highest derivative that appears in the equation
First order DE 
𝑑𝑦 𝑑𝑥 =𝑐𝑜𝑠𝑥
(𝑥2 +𝑦2)𝑑𝑥 −2𝑥𝑦𝑑𝑦 = 0
𝑥𝜕𝑓 𝜕𝑥 +𝑦𝜕𝑓 𝜕𝑦 +𝑤=0
Second order DE 
𝑑2𝑦 𝑑𝑥2 +𝑘2𝑦 =0
𝜕2𝑉 𝜕𝑥2
(𝑑2𝑤 𝑑𝑥2 +𝜕2𝑉 𝜕𝑦2 3 ) =0 −𝑥𝑦𝑑𝑤 𝑑𝑥 +𝑤=0
Linear ODE 
Of the form 𝑛(𝑥)��𝑛𝑦 𝑑𝑥𝑛 +𝑎𝑛−1(𝑥)𝑑𝑛−1𝑦 𝑑𝑥𝑛−1 +⋯+𝑎1(𝑥)𝑑𝑦 𝑑𝑥 +𝑎0(𝑥)𝑦 = 𝑔(𝑥)
Nonlinear ODE 
One that is not linear or not in the form of 𝑛(𝑥)𝑑𝑛𝑦 𝑑𝑥𝑛 +𝑎𝑛−1(𝑥)𝑑𝑛−1𝑦 𝑑𝑥𝑛−1 +⋯+𝑎1(𝑥)𝑑𝑦 𝑑𝑥 +𝑎0(𝑥)𝑦 = 𝑔(𝑥)
Linear DE 
𝑑𝑦 𝑑𝑥 =𝑐𝑜𝑠𝑥
𝑑2𝑦 𝑑𝑥2 +��2𝑦 =0
Nonlinear DE 
(𝑥2 +𝑦2)𝑑𝑥 −2𝑥��𝑑𝑦 = 0
(𝑑2𝑤 𝑑𝑥2 +𝜕2𝑉 𝜕��2 3 ) =0 −𝑥��𝑑𝑤 𝑑𝑥 +𝑤=0