Differential Equations

Chapter Review

THINGS TO KNOW

16.1 Classification of Ordinary Differential Equations

Ordinary differential equation (p. 1059)
Order of a differential equation: the highest-order derivative of \(y\) in the equation (p. 1059)
Degree of a differential equation: the exponent of the highest-order derivative in the equation (p. 1059)
Linear differential equation of order \(n\) (p. 1059)

16.2 Separation of Variables in First-Order Differential Equations

Separable first-order differential equations (p. 1062)
Steps for Solving a Separable Differential Equation (p. 1062)
The function \(f(x,y)\) is a homogeneous function of degree \(k\) in \(x\) and \(y\) if and only if for all \(t>0\), \(f(tx,ty)=t^{k}f(x,y),\) where \(k\) is some real number. (p. 1063)
A differential equation of the form \(M(x,y)\,dx+ N(x,y)\,dy=0\) is homogeneous if \(M\) and \(N\) are homogeneous functions of the same degree. (p. 1064)
Steps for Solving a Homogeneous First-Order Differential Equation (p. 1064)

16.3 Exact Differential Equations

Exact differential equation (p. 1073)
A differential equation of the form \(M(x,y)\,dx+ N(x,y)\,dy=0\) is an exact differential equation at every point \((x,y)\) in a simply connected open set \(R\) if and only if \(\dfrac{\partial M}{\partial y}=\dfrac{\partial N }{\partial x}\) for all \(( x,y) \) in \(R.\) (p. 1073)
Integrating factor (p. 1076)

16.4 First-Order Linear Differential Equations; Bernoulli Differential Equations

First-order linear differential equation: \(\dfrac{dy}{dx}+P(x)y=Q(x), \) where \(P\) and \(Q\) are continuous on their domains (p. 1078)
If \(Q( x) =0,\) the first-order linear differential equation is separable. (p. 1078)
If \(Q( x) \neq 0,\) use the integrating factor \(e^{\int \!P(x)\,dx}.\) (p. 1078)
Bernoulli equation: \(\dfrac{dy}{dx}+P(x)y=Q(x)y^{n},n\neq 0, n\neq 1,\) and \(P\) and \(Q\) are functions of \(x.\) (p. 1082)
Steps for Solving a Bernoulli Equation (p. 1082)

16.5 Power Series Methods (pp. 1089-1093)

Assumes a solution \(y=y(x)\) of a differential equation has a power series expansion of the form \(y=y( x) =\sum\limits_{k=0}^{\infty }a_{k}x^{k}.\) (p. 1089)

1095

OBJECTIVES

Section	You should be able to ...	Examples	Review Exercises
16.1	1 Classify ordinary differential equations (p. 1059)	1	1–4
	2 Verify the solution of an ordinary differential equation (p. 1060)	2	5–8
16.2	1 Solve a separable first-order differential equation (p. 1062)	1	13, 19, 29, 30, 32, 34
	2 Identify a homogeneous function of degree \(k\) (p. 1063)	2	9–12
	3 Use a change of variables to solve a homogeneous first-order differential equation (p. 1064)	3, 4	21, 26, 28, 33
	4 Solve applied problems (p. 1066)	5, 6	39, 41
16.3	1 Identify and solve an exact differential equation (p. 1074)	1–3	14, 17, 20
16.4	1 Solve a first-order linear differential equation (p. 1078)	1	15, 16, 18, 22, 25, 27, 31
	2 Solve applied problems involving first-order linear differential equations (p. 1079)	2, 3	40
	3 Find the general solution of a Bernoulli equation (p. 1082)	4	23, 24
	4 Solve applied problems (p. 1083)	5	42
16.5	1 Use power series to solve a linear differential equation (p. 1089)	1–4	35–38