9:19 4 .. .. Safari K AA Q N 1 of 5 16 Sat 17 next Mon Forum Grapher Calculus II : 17 Introduction to Differential Equations | du Live Notes Expand All PDF-beta Examples of differential equations - An equation that contains the 2 Slope fields - Differential equations, such as the ones illustrated in the previous 3 Growth and decay equation - A linear first differential equation given by f(t) = k*f(t), or by dy/dt = k y by letting y = f(t) is called the exponential growth and dec... 4 Guessing a solution - Let dy/dt = 0.5 y. This means that at any value t the derivative of some function y = f(t) is one half of the function itself. Graphically, ... 5 The general solution of dy/dy=ky. - All differentiable functions whose instantaneous rates of change are proportional to their values must exponential functions.9:194 4 Safari (SE @AAQEI III. ? E]. 20f5 This is proved... - Population Growth model - The equation dy/dt = k y often models a population growth when multiplying is not offset by limited resources such as space, nutrition or na... 7 Carbon Dating - How to estimate the age of organic remains from archaeological sites - The radioactive decay of carbonA(14)C follows the exponential decay equation: dy/dt = k y, where y is the number ofA(14)C atoms at time t an... . Newton's Law of Cooling - Assume that are are in a room with a constant temperature A = 21"(circ). You leave a hot cup of tea on the table. The initial temperature of... 9 Interest compounded continuously - ...illustration... Let P(t) be the value of an account at time t . The rate of change of P(t) divided by the value P(t) is called the relati... . Interest compounded several times - The term compound interest refers to adding the interest to the principal and then taking the next interest on the principal and the previou... 11 The limit formula for the number e=2.71... - In general if the initial amount P(O) is subjected to an interest of 100 \\%compounded n times a year, the value after one year is P(l) = P(O... . Checking solutions of higher order differential equations - An equation that has the second or higher order derivative of an unknown function is called a higher order differential equation . Physics g... 13 The general solution of the harmonic oscillator equation - The equation m[d"2y]/[dt"2] =-k y is often times called a simple harmonic oscillator equation. A mass m experiences a single force m[d"2y]/[... 8. . '10. g. j Describe in a short paragraph a phenomenon around you that manifests exponential growth or decay (other than a Guinness beer). When you share your example, ensure that no student has discribed an identical example earlier on the forum. Suppose that we can drill a tunnel connecting San Francisco and Paris. Traveling though it will be Gravistar, a frictionless train moving only due to the pull of gravity. How long will travel by Gravistar take? Will it be faster than the direct ight between San Franisco and Paris? San HanA Paris Let x denote the position of an object of mass m on the axis joining San Francisco and Paris. We set zero to be the midpoint between the two cities. From physics one can show that such an object is subjected to the effective gravitational force F = k:c where k = m - 1.55 - 104%" and F = mg according to the Newton's second Law of motion. Write down the differential equation that relates the position so and time t of an object in the tunnel. Then assume that at time t = 0 the object is let go in the tunnel from San Francisco. Use the the theorem from last section to calculate the time it will take Gravistar to reach Paris. Could two solutions to rst order differential equation Edit! 2 F(t, y) colide? Would there be a need for speed limit if we always drove along a solution to the same differential equation