Ch. 7 – Differential Equations and Mathematical Modeling 7.4 Solving Differential Equations

Separable Differential Equation = a diff. eqn. in the form –To solve this equation, we must separate the variables… –…and antidifferentiate each side individually. Ex: Solve for y if y(0) = -1: –Separate the variables (x’s on one side, y’s on the other) –Now antidifferentiate each side with respect to the correct variable… –You only need a C on one side because “some constant” minus “some constant” will just be “some constant” –Apply the initial condition and solve!

Ex: Solve for y if y(4) = -3: –Separate the variables (x’s on one side, y’s on the other) –Now antidifferentiate each side with respect to the correct variable… –Apply the initial condition and solve!

Ex: Solve for y if y(3) = 1: –Separate the variables (x’s on one side, y’s on the other) –Now antidifferentiate each side with respect to the correct variable… –Apply the initial condition and solve! –This equation can also be written as follows: –In general, when solving a differential eqn. of the form dy/dt = ky, where k is some growth constant, then the solution to the diff. eqn. is in the form of A is the initial value of y when t = 0

Compound Interest Formula: –A (t) = amount at time t –A 0 = initial amount t=0) –r = interest rate –n (book uses k) = number of times per year interest is compounded Continuously Compounded Interest Formula: –For exponential decay, r is negative Newton’s Law of Cooling: –T = temperature of an object at time t –T s = surrounding temperature –T 0 = initial temperature t=0) –k = cooling constant