Section 1.6
by C. Henry Edwards,
David E. Penney,
David T. Calvis,
Differential Equations and Linear Algebra, 4th Edition
- Differential Equations & Linear Algebra
- Contents
- Application Modules
- Preface
- 1 First-Order Differential Equations
- 2 Mathematical Models and Numerical Methods
- 2.1 Population Models
- 2.2 Equilibrium Solutions and Stability
- 2.3 Acceleration–Velocity Models Acceleration–Velocity Models
- 2.4 Numerical Approximation: Euler’s Method Numerical Approximation: Euler’s Method
- 2.5 A Closer Look at the Euler Method
- An Improvement in Euler’s Method
- 2.6 The Runge–Kutta Method The Runge–Kutta Method
- 3 Linear Systems and Matrices
- 4 Vector Spaces
- 5 Higher-Order Linear Differential Equations
- 5.1 Introduction: Second-Order Linear Equations
- 5.2 General Solutions of Linear Equations
- 5.3 Homogeneous Equations with Constant Coefficients
- 5.4 Mechanical Vibrations
- 5.5 Nonhomogeneous Equations and Undetermined Coefficients
- 5.6 Forced Oscillations and Resonance
- 6 Eigenvalues and Eigenvectors
- 7 Linear Systems of Differential Equations
- 7.1 First-Order Systems and Applications
- 7.2 Matrices and Linear Systems
- 7.3 The Eigenvalue Method for Linear Systems
- 7.4 A Gallery of Solution Curves of Linear Systems
- Systems of Dimension n=2
- Real Eigenvalues
- Saddle Points
- Nodes: Sinks and Sources
- Zero Eigenvalues and Straight-Line Solutions
- Repeated Eigenvalues; Proper and Improper Nodes
- The Special Case of a Repeated Zero Eigenvalue
- Complex Conjugate Eigenvalues and Eigenvectors
- Pure Imaginary Eigenvalues: Centers and Elliptical Orbits
- Complex Eigenvalues: Spiral Sinks and Sources
- A 3-Dimensional Example
- 7.4 Problems
- 7.4 Application Dynamic Phase Plane Graphics
- 7.5 Second-Order Systems and Mechanical Applications*
- 7.6 Multiple Eigenvalue Solutions
- 7.7 Numerical Methods for Systems
- 7.7 Application Comets and Spacecraft
- 8 Matrix Exponential Methods
- 9 Nonlinear Systems and Phenomena
- 9.1 Stability and the Phase Plane
- 9.2 Linear and Almost Linear Systems
- 9.3 Ecological Models: Predators and Competitors
- 9.4 Nonlinear Mechanical Systems
- 10 Laplace Transform Methods
- 11 Power Series Methods
- References for Further Study
- APPENDIX A Existence and Uniqueness of Solutions
- APPENDIX B Theory of Determinants
- Answers to Selected Problems
- Index
Section 1.6
1.
x2−2xy−y2=C 2.
y2=x2(ln x+C) 3.
y(x)=x(C+ln |x|)2 4.
2 tan−1(y/x)−ln(y2/x2+1)=2 ln x+C 5.
ln |xy|=C+xy−1 6.
2y ln y=x+Cy 7.
y3=3x3(C+ln|x|) 8.
y=−x ln(C−ln x) 9.
y(x)=x/(C− ln|x|) 10.
x2+xy2=Cx6 11.
y=C(x2+y2) 12.
4x2+y2=x2(ln x+C)2 13.
y+x2+y2−−−−−−√=Cx2 14.
x−x2+y2−−−−−−√=C 15.
x2(2xy+y2)=C 16.
x=2x+y+1−−−−−−−−√−2 ln(1+x+y+1−−−−−−−−√)+C 17.
y(x)=−4x+2 tan(2x+C) 18.
y=ln(x+y+1)+C 19.
y2=x/(2+Cx5) 20.
y3=3+Ce−3x2 21.
y2=1/(Ce−2x−1) 22.
y3=7x/(7Cx7+15) 23.
y(x)=(x+Cx2)−3 24.
y2=e2x/(C+ln x) 25.
2x3y3=31+x4−−−−−√+C 26.
y3=e−x(x+C) 27.
y(x)=(x4+Cx)1/3 28.
y=ln(Cx2+x2e2x) 29.
sin2y=4x2+Cx 30.
x2−2xey−e2y=C 31.
x2+3xy+y2=C 32.
2x2−xy+3y2=C 33.
x3+2xy2+2y3=C 34.
x3+x2y2+y4=C 35.
3x4+4y3+12y ln x=C 36.
x+exy+y2=C 37.
sin x+x ln y+ey=C 38.
x2+2x tan−1y+ln(1+y2)=C 39.
5x3y3+5xy4+y5=C 40.
exsin y+xtan y=C 41.
x2y−1+y2x−3+2y1/2=C 42.
xy−2/3+x−3/2y=C 43.
y(x)=Ax2+B 44.
x(y)=Ay2+B 45.
y(x)=A cos 2x+B sin 2x 46.
y(x)=x2+A ln x+B 47.
y(x)=A−ln |x+B| 48.
y(x)=ln x+Ax−2+B 49.
y(x)=±(A+Bex)1/2 50.
y(x)=ln|sec(x+A)|−12x2+B 51.
x(y)=−13(y3+Ay+B) 52.
Ay2−(Ax+B)2=1 53.
y(x)=A tan(Ax+B) 54.
Ay2(B−x)=1 58.
y=exp(x2+C/x2) 59.
x2−2xy−y2−2x−6y=C 60.
(x+3y+3)5=C(y−x−5) 61.
x=tan(x−y)+sec(x−y)+C 64.
y(x)=x+e−x2[C+12π−−√ erf(x)]−1 65.
y(x)=x+(C−x)−1 69. Approximately 3.68 mi
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