MT1710 - 2018/19: WEEK 9 Chapter IV: Ordinary Differential Equations (ODE). Section 4.1. - Introduction. Definition 4.1: A differential equation is any equation which contains derivatives. An ordinary differential equation contains only total derivatives, as opposed to partial derivatives. Definition 4.2: The order of an ordinary differential equation (ODE) is the order of the highest derivative that appears in that equation. The degree of an ordinary differential equation is the power of the highest derivative when the equation has been made rational with respect to the derivatives. Example 4.1 - Find the order and degree of each of the following differential equa- tions: (a) d3 y dx3 + px dx2 d2 y + tan y Ax dy = 4. (b) dy dx = 3- . 1+ 27 32 dx2 dy dx (c) x y3 (1 + x4) 2 General form of a first order and first degree ODE: dy Q(x, y) + P(x, y) = 0, dx where P and Q are functions of x and y. Definition 4.3: A linear ODE is such that it can be written in the form d" y dan-1 d"-1y + . + ?1 (c) dx dy + ao (x) y = g (x) , an (x) dxn + an-1 (x) for ai(x) and g(x) continuous functions. n, n - 1, ··· are the order of the successive derivatives. :
N.B .- In linear differential equations the function y(x) and its derivatives occur only to the first power: y(t) does not occur in products or in arguments of transcenden- tal functions. Only the function y(t) and its derivatives are used to determine if a differential equation is linear. General form of a first order linear Ordinary Differential Equation: dy +p(x)y= g(x) , dx where p and g are given continuous functions of x. General form of a second order linear Ordinary Differential Equation: d2y dx2 + p (x) dy dx + q (x) y = f (x) , where p, q and f are given continuous functions of x. Section 4.2. - Some first order Ordinary Differential Equations. Section 4.2.1. - Trivial type. e.g. dx dy = sin (2x) Section 4.2.2. - Separable Equations. Definition 4.4: A first order differential equation that can be written in the form M(x) + N (y) dy _ 0 dx is said to be separable. Example 4.2 Solve the differential equations: dy (a) dx x 1 + y2' 2 (b) x + ye- dy dx =0, y(0) = 1.
Section 4.2.3. - Exact Equations. Definition 4.5: An equation of the form Q(x, y) dy + P(x, y) = 0 is said to be exact if it can be written in the form d dx [f(x, y)] = 0, for some differentiable function f. Theorem 4.1: A necessary and sufficient condition for the equation to be exact is that dy dx + P(x,y) = 0 Q(x, y) OP aQ . ?? Examples 4.3 Solve the following differential equations: (i) x dy + y = 0 , (ii) 2xy3 +