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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