Title 86 Degree and Order.pdf ✅

MSTC 85: Degree and Order 1. Types of Differential Equations Differential equations are equations that involve a differential or derivative of a variable. Consider the equation y′ +Ay′′ = Bx Knowing that y′ = dy dx , the quantities of the above equation can be defined as follows: y = dependent variable x = independent variable A, B = constants In the above example, the equation has a derivative. However, a differential equation may also have just differentials. For example, (x + y)dx − xy dy = 0 This equation contains the differentials dx and dy. Differential equations can be classified as either ordinary or partial. • Ordinary Differential Equations (ODE). Differential equations containing only two variables. Examples: y ′ + Ay ′′ = Bx (x + y)dx − xy dy = 0 • Partial Differential Equations (PDE). Differential equations with three or more variables, thus including partial derivatives. Examples: x ∂z ∂x + y ∂z ∂y = 5 2. Parts of Differential Equations • Order. The order of a differential equation is the order of the highest-ordered derivative involved. • Degree. The degree of a differential equation is the power of the highest-ordered derivative involved. • Solution. A solution to a differential equation is a function that satisfies it. For example, (y ′′′) 2 + (y ′ ) 4 = x In this equation, the order is 3, and the degree is 2. y (5) + y 6 = 0 In this equation, the order is 5, and the degree is 1. 3. Types of Solutions to Differential Equations. • General Solution. It is a complete solution that includes all the possible solutions to a differential equation. It also involves arbitrary constants. • Particular Solution. If one or more of the arbitrary constants of the general solution are fixed to a particular value, the solution is called a particular solution.