MT1710 - 2018/19: WEEK 10 Section 4.2.5. - Bernoulli's Differential Equation. Bernoulli's differential equation has the form, dy + p(x)y = q(x)y" n = 0, n = 1. dx It is a non-linear differential equations with a known exact solution. Details given in the lecture. Divide through by y": Substitute v = y1-2 so dv = (1 -n)y-n dx dy dx y" dx 1 dy + p(x)y1-" = q(x) and hence 1 dv 1 - n dx + p(x)v = q(x) a linear equation. Example 4.6 (Exam May 2015) Find the general solution of the differential equation 2 (1 - x2) 9 dx dy + x y = y3, x > 1. Section 4.3: Second-Order Linear Differential Equations. Section 4.3.1. - Linear Differential Equations with Constant Coefficients. Definition 4.8: The general nth-order linear differential equation with constant coef- ficients has the form a0 dxn d + @1 dn-1y den-1 + ... + any = f(x) for constants ao # 0, a1, ... , an. When f(x) = 0, the equation is said to be homogeneous and we refer to it as the nth-order complementary equation, a0 d" y den + @1 dxn-1 dn-1y + . + any =0.
Specialising to the second order equation, we then write the 2nd-order linear differential equation with constant coefficients as d ao- dx2 dy + 01 px + @2y = f (x) . for constants ao # 0, a1 and a2. When f(x) = 0, the equation is homogeneous and we refer to the 2nd order complementary equation, ao ?2 y dx2 + @1- dx dy + @2y = 0. Theorem 4.2: Given a second-order linear differential equation with constant co- efficients such that y1(x) and y2(x) are independent solutions of the complementary equation, i.e., solutions satisfying dy1 y1 dx dy2 - y2 dx ¥0, then every solution yc(x) of the complementary equation may be written in the form, yc(x)= C1y1(x) + C2y2(x), by suitable choice of the arbitrary constants c1 and C2. In general, the solution of a complementary nth-order equation will contain n arbitrary constants. Definition 4.9: Any solution yp(x) of the linear differential equation with constant coefficients, containing no arbitrary constants, is said to be a particular integral. The general solution of the complementary equation, involving n arbitrary constants, is said to be the complementary function. ..
Theorem 4.3: Let yp(x) be a particular integral of the linear equation, with constant coefficients, ao dx2 d2 y dy + a1 + @2y =f(x) dx and yc(x) be the complementary function. Then the general solution of the given equation is y(x) = yc(x) + yp(x) . How to find the complementary function? Example 4.7 Consider the following differential equation 2 + 5- dx dy +2y = 5+2x dx2 The solution of a general linear differential equation can now be divided into two stages - finding the complementary function; - finding a particular integral. Section 4.3.2. - The Complementary Function. Working with the second-order differential equation with constant coefficients, ao ?2 y dr.2 +