chapter13Problems - Chapter 13 More Dierential Equations...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 13 More Differential Equations 13.1 Consider the differential equation dy dt = a - by where a , b are constants. (a) Show that the function y ( t ) = a b - Ce - bt satisfies the above differential equation for any constant C . (b) Show that by setting C = a b - y 0 we also satisfy the initial condition y (0) = y 0 . Remark: You have now shown that the function y ( t ) = parenleftBig y 0 - a b parenrightBig e - bt + a b is a solution to the initial value problem (i.e differential equation plus initial condition) dy dt = a - by, y (0) = y 0 . 13.2 For each of the following, show the given function y is a solution to the given differential equation. (a) t · dy dt = 3 y , y = 2 t 3 . (b) d 2 y dt 2 + y = 0, y = - 2 sin t + 3 cos t . (c) d 2 y dt 2 - 2 dy dt + y = 6 e t , y = 3 t 2 e t . v.2005.1 - September 4, 2009 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Math 102 Problems Chapter 13 13.3 Show the function determined by the equation 2 x 2 + xy - y 2 = C , where C is a constant and 2 y negationslash = x , is a solution to the differential equation ( x - 2 y ) dy dx = - 4 x - y . 13.4 Find the constant C that satisfies the given initial conditions. (a) 2 x 2 - 3 y 2 = C , y | x =0 = 2. (b) y = C 1 e 5 t + C 2 te 5 t , y | t =0 = 1 and dy dt | t =0 = 0. (c) y = C 1 cos( t - C 2 ), y | t = π 2 = 0 and dy dt | t = π 2 = 1. 13.5 Friction and terminal velocity The velocity of a falling object changes due to the acceleration of gravity, but friction has an effect of slowing down this acceleration. The differential equation satisfied by the velocity v ( t ) of the falling object is dv dt = g - kv where g is acceleration due to gravity and k is a constant that represents the effect of friction. An object is dropped from rest from a plane. (a) Find the function v ( t ) that represents its velocity over time. (b) What happens to the velocity after the object has been falling for a long time (but before it has hit the ground)? 13.6 Alcohol level Alcohol enters the blood stream at a constant rate k gm per unit time during a drinking session. The liver gradually converts the alcohol to other, non-toxic byproducts. The rate of conversion per unit time is proportional to the current blood alcohol level, so that the differential equation satisfied by the blood alcohol level is dc dt = k - sc where k , s are positive constants. Suppose initially there is no alcohol in the blood. Find the blood alcohol level c ( t ) as a function of time from t = 0, when the drinking started.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern