6_pdfsam_math 54 differential equation solutions odd

6_pdfsam_math 54 differential equation solutions odd - x...

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Chapter 1 11. This equation contains partial derivatives, thus it is a PDE. Because the highest order deriva- tive is a second order partial derivative, the equation is a second order equation. The terms ∂N/∂t and ∂N/∂r show that the independent variables are t and r and the dependent variable is N . 13. Since the rate of change of a quantity means its derivative, denoting the coefficient propor- tionality between dp/dt and p ( t )by k ( k> 0), we get dp dt = kp. 15. In this problem, T M (coFee is hotter than the air), and T is a decreasing function of t , that is dT/dt 0. Thus dT dt = k ( M T ) , where k> 0 is the proportionality constant. 17. In classical physics, the instantaneous acceleration, a , of an object moving in a straight line is given by the second derivative of distance,
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Unformatted text preview: x , with respect to time, t ; that is d 2 x dt 2 = a. Integrating both sides with respect to t and using the given fact that a is constant we obtain dx dt = at + C. (1.1) The instantaneous velocity, v , of an object is given by the ±rst derivative of distance, x , with respect to time, t . At the beginning of the race, t = 0, both racers have zero velocity. Therefore we have C = 0. Integrating equation (1.1) with respect to t we obtain x = 1 2 at 2 + C 1 . ²or this problem we will use the starting position for both competitors to be x = 0 at t = 0. Therefore, we have C 1 = 0. This gives us a general equation used for both racers as x = 1 2 at 2 or t = r 2 x a , 2...
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