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Calculus Volume 3

Key Concepts

Calculus Volume 3Key Concepts
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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

7.1 Second-Order Linear Equations

  • Second-order differential equations can be classified as linear or nonlinear, homogeneous or nonhomogeneous.
  • To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions. If y1(x)y1(x) and y2(x)y2(x) are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by
    y(x)=c1y1(x)+c2y2(x).y(x)=c1y1(x)+c2y2(x).
  • To solve homogeneous second-order differential equations with constant coefficients, find the roots of the characteristic equation. The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots.
  • Initial conditions or boundary conditions can then be used to find the specific solution to a differential equation that satisfies those conditions, except when there is no solution or infinitely many solutions.

7.2 Nonhomogeneous Linear Equations

  • To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation.
  • Let yp(x)yp(x) be any particular solution to the nonhomogeneous linear differential equation
    a2(x)y+a1(x)y+a0(x)y=r(x),a2(x)y+a1(x)y+a0(x)y=r(x),

    and let c1y1(x)+c2y2(x)c1y1(x)+c2y2(x) denote the general solution to the complementary equation. Then, the general solution to the nonhomogeneous equation is given by
    y(x)=c1y1(x)+c2y2(x)+yp(x).y(x)=c1y1(x)+c2y2(x)+yp(x).
  • When r(x)r(x) is a combination of polynomials, exponential functions, sines, and cosines, use the method of undetermined coefficients to find the particular solution. To use this method, assume a solution in the same form as r(x),r(x), multiplying by x as necessary until the assumed solution is linearly independent of the general solution to the complementary equation. Then, substitute the assumed solution into the differential equation to find values for the coefficients.
  • When r(x)r(x) is not a combination of polynomials, exponential functions, or sines and cosines, use the method of variation of parameters to find the particular solution. This method involves using Cramer’s rule or another suitable technique to find functions u(x)u(x) and v(x)v(x) satisfying
    uy1+vy2=0uy1+vy2=r(x).uy1+vy2=0uy1+vy2=r(x).

    Then, yp(x)=u(x)y1(x)+v(x)y2(x)yp(x)=u(x)y1(x)+v(x)y2(x) is a particular solution to the differential equation.

7.3 Applications

  • Second-order constant-coefficient differential equations can be used to model spring-mass systems.
  • An examination of the forces on a spring-mass system results in a differential equation of the form
    mx+bx+kx=f(t),mx+bx+kx=f(t),

    where mm represents the mass, bb is the coefficient of the damping force, kk is the spring constant, and f(t)f(t) represents any net external forces on the system.
  • If b=0,b=0, there is no damping force acting on the system, and simple harmonic motion results. If b0,b0, the behavior of the system depends on whether b24mk>0,b24mk>0, b24mk=0,b24mk=0, or b24mk<0.b24mk<0.
  • If b24mk>0,b24mk>0, the system is overdamped and does not exhibit oscillatory behavior.
  • If b24mk=0,b24mk=0, the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior.
  • If b24mk<0,b24mk<0, the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time.
  • If f(t)0,f(t)0, the solution to the differential equation is the sum of a transient solution and a steady-state solution. The steady-state solution governs the long-term behavior of the system.
  • The charge on the capacitor in an RLC series circuit can also be modeled with a second-order constant-coefficient differential equation of the form
    Ld2qdt2+Rdqdt+1Cq=E(t),Ld2qdt2+Rdqdt+1Cq=E(t),

    where L is the inductance, R is the resistance, C is the capacitance, and E(t)E(t) is the voltage source.

7.4 Series Solutions of Differential Equations

  • Power series representations of functions can sometimes be used to find solutions to differential equations.
  • Differentiate the power series term by term and substitute into the differential equation to find relationships between the power series coefficients.
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