[...]... ∂x (1.11) Exact equations are solved in Chapter Two (where a more precise definition of exactness is given) Chapter 2 Solutions of First-Order Differential Equations In This Chapter: ✔ ✔ ✔ ✔ ✔ ✔ Separable Equations Homogeneous Equations Exact Equations Linear Equations Bernoulli Equations Solved Problems Separable Equations General Solution The solution to the first-order separable differential equation... differential equations does “homogeneous” have the meaning defined above Separable Equations Consider a differential equation in differential form (1.7) If M(x,y) = A(x) (a function only of x) and N(x,y) = B(y) (a function only of y), the differential equation is separable, or has its variables separated Separable equations are solved in Chapter Two Exact Equations A differential equation in differential. .. differential equation is linear First-order linear differential equations can always be expressed as y ′ + p( x ) y = q( x ) (1.8) Linear equations are solved in Chapter Two Bernoulli Equations A Bernoulli differential equation is an equation of the form y ′ + p( x ) y = q( x ) y n (1.9) where n denotes a real number When n = 1 or n = 0, a Bernoulli equation reduces to a linear equation Bernoulli equations. .. Homogeneous Equations A differential equation in standard form (1.6) is homogeneous if f (tx, ty) = f ( x, y) (1.10) for every real number t Homogeneous equations are solved in Chapter Two CHAPTER 1: Basic Concepts and Classification 7 Note! In the general framework of differential equations, the word “homogeneous” has an entirely different meaning (see Chapter Four) Only in the context of first-order differential. .. problem is a function y(x) that both solves the differential equation and satisfies all given subsidiary conditions Standard and Differential Forms Standard form for a first-order differential equation in the unknown function y(x) is y ′ = f ( x, y) (1.6) where the derivative y ′ appears only on the left side of 1.6 Many, but not all, first-order differential equations can be written in standard form by... f(x,y) equal to the right side of the resulting equation 6 DIFFERENTIAL EQUATIONS The right side of 1.6 can always be written as a quotient of two other functions M(x,y) and −N(x,y) Then 1.6 becomes dy / dx = M ( x, y) / − N ( x, y), which is equivalent to the differential form M ( x, y)dx + N ( x, y)dy = 0 (1.7) Linear Equations Consider a differential equation in standard form 1.6 If f(x,y) can be... the solution to the given differential equation as y = x ln | kx | 16 DIFFERENTIAL EQUATIONS Solved Problem 2.3 Solve 2 xydx + (1 + x 2 )dy = 0 This equation has the form of Equation 2.11 with M(x, y) = 2xy and N(x, y) = 1 + x2 Since ∂M / ∂y = ∂N / ∂x = 2 x, the differential equation is exact Because this equation is exact, we now determine a function g(x, y) that satisfies Equations 2.14 and 2.15 Substituting... 28 DIFFERENTIAL EQUATIONS The solution of this linear equation is Q = ce−t /20 (3.20) At t = 0, we are given that Q = a = 20 Substituting these values into 3.20, we find that c = 20, so that 3.20 can be rewritten as Q = 20e−t /20 Note that as t → ∞, Q → 0 as it should, since only fresh water is being added Chapter 4 Linear Differential Equations: Theory of Solutions In This Chapter: ✔ ✔ ✔ ✔ Linear Differential. .. only fresh water is being added Chapter 4 Linear Differential Equations: Theory of Solutions In This Chapter: ✔ ✔ ✔ ✔ Linear Differential Equations Linearly Independent Solutions The Wronskian Nonhomogeneous Equations Linear Differential Equations An nth-order linear differential equation has the form bn ( x ) y ( n ) + bn −1 ( x ) y ( n −1) + L + b1 ( x ) y ′ + b0 ( x ) y = g( x ) (4.1) where g(x) and... (See Problem 2.7) Bernoulli Equations A Bernoulli differential equation has the form y ′ + p( x ) y = q( x ) y n (2.24) where n is a real number The substitution z = y1− n (2.25) transforms 2.24 into a linear differential equation in the unknown function z(x) (See Problem 2.8) Solved Problems Solved Problem 2.1 Solve dy x 2 + 2 = dx y This equation may be rewritten in the differential form ( x 2 + 2)dx . the work will meet your requirements or that its operation will be unin- terrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccu- racy,. 0-0 7-1 4284 6-1 The material in this eBook also appears in the print version of this title: 0-0 7-1 40967-X All trademarks are trademarks of their respective owners. Rather than put a trademark. publish or sub- license the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own non- commercial and personal use; any other use of the work is strictly prohibited.