By Thomas Craig
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Additional resources for A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients
Y − 2t y = 1; y = et t 2 2 e−s ds + et 2 0 In each of Problems 15 through 18 determine the values of r for which the given differential equation has solutions of the form y = er t . 15. y + 2y = 0 16. y − y = 0 17. y + y − 6y = 0 18. y − 3y + 2y = 0 In each of Problems 19 and 20 determine the values of r for which the given differential equation has solutions of the form y = t r for t > 0. 19. t 2 y + 4t y + 2y = 0 20. t 2 y − 4t y + 4y = 0 In each of Problems 21 through 24 determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear.
To confirm this, observe that y1 (t) = − sin t and y1 (t) = − cos t; then it follows that y1 (t) + y1 (t) = 0. In the same way you can easily show that y2 (t) = sin t is also a solution of Eq. (17). Of course, this does not constitute a satisfactory way to solve most differential equations because there are far too many possible functions for you to have a good chance of finding the correct one by a random choice. Nevertheless, it is important to realize that you can verify whether any proposed solution is correct by substituting it into the differential equation.
B) Solve the problem in part (a). (c) You have invited several dozen friends to a pool party that is scheduled to begin in 4 hr. 02 g/gal. Is your filtering system capable of reducing the dye concentration to this level within 4 hr? 02 g/gal. 02 g/gal within 4 hr. 3 Classification of Differential Equations The main purpose of this book is to discuss some of the properties of solutions of differential equations, and to describe some of the methods that have proved effective in finding solutions, or in some cases approximating them.
A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients by Thomas Craig