Ordinary Differential Equations Introduction A differential equation is an equation involving an unknown function and its derivatives. A differential equation is called an ordinary differential equation (ODE) if the unknown function depends on only a single independent variable. If the unknown function depends on more than one independent variables, the differential equation is called a partial differential equation (PDE). Here are some examples of differential equations: 1) 2) 3)
đ?‘‘đ?‘Ś
3 đ?‘‘đ?‘Ľ = 2đ?‘Ś − 3 đ?‘‘2 đ?‘Ś đ?‘‘đ?‘Ľ 2 đ?‘‘2 đ?‘Ś đ?‘‘đ?‘Ľ 2
; ODE
đ?‘‘đ?‘Ś
+ đ?‘‘đ?‘Ľ + 5đ?‘Ś = đ?‘Ľ 2 ; ODE đ?‘‘đ?‘Ś
− 2 đ?‘‘đ?‘Ą = 0
; PDE
where đ?‘Ľ is known as an independent variable and đ?‘Ś is a dependent variable or an unknown function. A differential equation is called an ordinary differential equation (ODE) if the unknown function depends on only a single independent variable. If the unknown function depends on more than one independent variables, the differential equation is called a partial differential equation (PDE). The order of a differential equation is the order of the highest derivative in the equation. The degree of a differential equation is the degree of the highest order derivative which appears in the equation. The following table shows examples of ordinary differential equations with different order and degree: Ordinary differential equation dy  e x2 y dx d2y dy ď€5 0 2 dx dx
Order 1
Degree 1
2
1
2
3
1
ďƒŚ d 2 y ďƒś dy ďƒ§ďƒ§ 2 ďƒˇďƒˇ  ď€ 3y  x ďƒ¨ dx ďƒ¸ dx
2
3
3 x  1ď€ŠďƒŚďƒ§ dy ďƒśďƒˇ  d y3  sin x dx ďƒ¨ dx ďƒ¸ 3
1