ode_dreamin_ch2_5-04-07[7]

ode_dreamin_ch2_5-04-07[7] - ODE DREAMIN CHAPTER 2...

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ODE DREAMIN’ CHAPTER 2 ADDITIONAL INTRODUCTORY MATERIAL (5-04-07) by Doug Jones ode_dreamin_ch2_5-04-07[7].doc Page 1 of 10 Here is some practice at the important skill of verifying that a given function is a solution of a ODE . Verification is how you check your answers, and if you can’t check your answers, how do you know if they are right? And since I’ve had some extra time on my hands during the short break between semesters, I’ve also included some review of important Calc II techniques we’ll be using in this course. Enjoy. I. Verify that () 2 t te φ = is a solution of 91 40 yy y ′′ + += . Verification : [1] 1 . () ( ) ( ) 22 2 24 tt t t e t e φφ −− =⇒ = −⇒ = [2]. Subs. [1] into the LHS of the ODE and simplify: ( ) ( )() 2 2 9 14 9 14 4 9 2 14 41 81 4 0 subs t t y e e e e ++ →+ + + =−+ = [3]. Thus, we see that the given function is in fact a solution to the ODE. II. Verify that 2 t = is a solution of 2 dy ty dt = on the half-plane 0 y > , given that y is a function of t. Verification : [1]. () () 2 t t e = [2]. Subs. [1] into the LHS of the ODE and simplify: 2 subs t dy d te t dt dt ⎯⎯⎯ →== [3]. Therefore, 2 d t dt = , and it follows (by def) that 2 t = is a solution of the given ODE. 1 Let’s get this “arrow” notation straight right at the outset! Misuse of the “arrow” ( ) is one of my pet-peeves! Technically, the arrow means “implies.” This means that the statement (or equation) that comes right before the arrow (the “antecedent”) is a sufficient condition, in-and-of itself, to imply the statement (or equation) that comes right after the arrow (the “consequent”). In writing math, we bend the rule just a little and use the arrow to mean that the antecedent plus possibly some other obvious facts together imply the consequent…. But in any case , the arrow means implies . It does not mean “equals.” ( = ) means “equals.” ( ) means “implies.” Don’t confuse them!
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ode_dreamin_ch2_5-04-07[7].doc Page 2 of 10 III. Verify that () ( ) 3sinh 2 tt φ = is a solution of 2 2 40 dx x dt = . Verification : [1]. () ( ) ( ) ( ) ( )( ) 3sinh 2 6cosh 2 12sinh 2 t t t t φφ ′′ =⇒ = = [2]. Subs. [1] into the LHS of the ODE and simplify: () () () 22 4 4 12sinh 2 4 3sinh 2 0 subs d xt t dt dt −⎯ →− = = [3]. Therefore, 2 2 d dt −= , and it follows (by def) that ( ) = is a solution of the given ODE. IV. Here is an “INSIGHT” for you to consider. For example, in the ODE above, 2 2 x dt = , think of the “zero” as a function ! V. Now let’s complicate things just a bit. Up until now, all the solutions we have verified have been explicit functions . However, in many, many problems you will be solving in this course, the solutions will be expressed as implicit functions . So I need to show you what such solutions look like and the slightly different technique used for verification.
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ode_dreamin_ch2_5-04-07[7] - ODE DREAMIN CHAPTER 2...

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