Differential Equations Workbook For Dummies
Language: English
Pages: 312
ISBN: 0470472014
Format: PDF / Kindle (mobi) / ePub
- Make sense of these difficult equations
- Improve your problem-solving skills
- Practice with clear, concise examples
- Score higher on standardized tests and exams
Get the confidence and the skills you need to master differential equations!
Need to know how to solve differential equations? This easy-to-follow, hands-on workbook helps you master the basic concepts and work through the types of problems you'll encounter in your coursework. You get valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every equation. You'll also memorize the most-common types of differential equations, see how to avoid common mistakes, get tips and tricks for advanced problems, improve your exam scores, and much more!
More than 100 Problems!
Detailed, fully worked-out solutions to problems
The inside scoop on first, second, and higher order differential equations
A wealth of advanced techniques, including power series
THE DUMMIES WORKBOOK WAY
Quick, refresher explanations
Step-by-step procedures
Hands-on practice exercises
Ample workspace to work out problems
Online Cheat Sheet
A dash of humor and fun
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........................................ 79 Chapter 4: Working with Linear Second Order Differential Equations ........................................ 81 Chapter 5: Tackling Nonhomogeneous Linear Second Order Differential Equations.............. 105 Chapter 6: Handling Homogeneous Linear Higher Order Differential Equations ..................... 129 Chapter 7: Taking On Nonhomogeneous Linear Higher Order Differential Equations............ 153 Part III: The Power Stuff: Advanced Techniques
2xe–x + e–x Chapter 4: Working with Linear Second Order Differential Equations 13. Solve this differential equation: What’s the solution to this equation? y" + 4y' + 4y = 0 y" + 10y' + 25y = 0 where where y(0) = 1 y(0) = 1 and and y'(0) = 0 y'(0) = 2 Solve It 15. 14. Solve this differential equation: Solve It 16. What’s the solution to this equation? y" + 8y' + 16y = 0 y" + 6y' + 9y = 0 where where y(0) = 2 y(0) = 4 and and y'(0) = 4 y'(0) = 4 Solve It Solve It
solution: y = yh + yp which can also be written as y = c1e–x + c2e–2x + 2x – 3 h Solve for the general solution to this equation: y" + 4y' + 3y = 3x + 10 Solution: y = c1e–x + c2e–3x + x + 2 1. Start by finding the homogeneous version of the original differential equation: y" + 4y' + 3y = 0 2. Assuming that the solution to the homogeneous equation is of the form y = erx means that when you substitute that solution into the differential equation, you get this characteristic equation: r 2 + 4r +
together to form your general solution, like so: y = c1e–x + c2xe–x +c3x2e–x j Solve this differential equation: y''' + 9y" + 27y' + 27y = 0 Solution: y = c1e–3x + c2xe–3x +c3x2e–3x 1. Because the problem features a third order differential equation with constant coefficients, try a solution of the form y = erx 2. Then plug the attempted solution into the equation: r 3erx + 9r 2erx + 27rerx + 27erx = 0 Chapter 6: Handling Homogeneous Linear Higher Order Differential Equations 3. Divide by erx
those covered in Chapter 1). That is, you may see something like this: But because the equations in this chapter are still considered first order, you can expect to see something along these lines: To restrict the form of this differential equation even more, say that M(x, y) is really just a function of x — that is, M(x). Similarly, say that N(x, y) is really just a function of y — that is, N(y). Combined, that gives you This differential equation is considered separable, because it can be
