Diffrential Equations, please help me solve this step by step.. Craig wants to l
ID: 1892151 • Letter: D
Question
Diffrential Equations, please help me solve this step by step..
Craig wants to lose 20 pounds before his high school reunion 100 days from now. His wife. Jenny, consults a diet book which says that he must consume 15 calories each day for every pound that he weighs in order to maintain his current weight. Jenny also notes that he will lose one pound of fat whenever he runs a cumulative deficit of 3500 calories. (In other words, if he consumes a total of 3500 calories less than what his body needs over any period of time, then he loses one pound.) Craig wants to go on a diet where he consumes the same number of calories each day. Suppose Craig consumes C calories per day and his current weight is W. Write an expression for his caloric deficit. What are the units on each term? Use your answer to (a) to write the differential equation that expresses the rate at which Craig loses weight as a function of his current weight. Craig supposes that since 20 pounds = 70,000 calories, he should consume 700 fewer calories each day for 100 days. Since Craig weighs 200 pounds today, he decides to consume 2300 calories per day (2300 = 15 times 200 - 700). Will his plan work? How much weight will Craig really lose in 100 days at 2300 cal/day? Determine the maximum number of calories that Craig should consume each day if he wants to weigh exactly 180 pounds 100 days from now. Is there an equilibrium solution? If so, what does it mean in the context of Craig's diet?Explanation / Answer
Differential Equation The first definition that we should cover should be that of differential equation. A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives. There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion. If an object of mass m is moving with acceleration a and being acted on with force F then Newton’s Second Law tells us. (1) To see that this is in fact a differential equation we need to rewrite it a little. First, remember that we can rewrite the acceleration, a, in one of two ways. (2) Where v is the velocity of the object and u is the position function of the object at any time t. We should also remember at this point that the force, F may also be a function of time, velocity, and/or position. So, with all these things in mind Newton’s Second Law can now be written as a differential equation in terms of either the velocity, v, or the position, u, of the object as follows. (3) (4) So, here is our first differential equation. We will see both forms of this in later chapters. Here are a few more examples of differential equations. (5) (6) (7) (8) (9) (10) Order The order of a differential equation is the largest derivative present in the differential equation. In the differential equations listed above (3) is a first order differential equation, (4), (5), (6), (8), and (9) are second order differential equations, (10) is a third order differential equation and (7) is a fourth order differential equation. Note that the order does not depend on whether or not you’ve got ordinary or partial derivatives in the differential equation. We will be looking almost exclusively at first and second order differential equations in these notes. As you will see most of the solution techniques for second order differential equations can be easily (and naturally) extended to higher order differential equations and we’ll discuss that idea later on. Ordinary and Partial Differential Equations A differential equation is called an ordinary differential equation, abbreviated by ode, if it has ordinary derivatives in it. Likewise, a differential equation is called a partial differential equation, abbreviated by pde, if it has differential derivatives in it. In the differential equations above (3) - (7) are ode’s and (8) - (10) are pde’s. The vast majority of these notes will deal with ode’s. The only exception to this will be the last chapter in which we’ll take a brief look at a common and basic solution technique for solving pde’s. Linear Differential Equations A linear differential equation is any differential equation that can be written in the following form. (11) The important thing to note about linear differential equations is that there are no products of the function, , and its derivatives and neither the function or its derivatives occur to any power other than the first power. The coefficients and can be zero or non-zero functions, constant or non-constant functions, linear or non-linear functions. Only the function, , and its derivatives are used in determining if a differential equation is linear. If a differential equation cannot be written in the form, (11) then it is called a non-linear differential equation. In (5) - (7) above only (6) is non-linear, the other two are linear differential equations. We can’t classify (3) and (4) since we do not know what form the function F has. These could be either linear or non-linear depending on F. Solution A solution to a differential equation on an interval is any function which satisfies the differential equation in question on the interval . It is important to note that solutions are often accompanied by intervals and these intervals can impart some important information about the solution. Consider the following example. Example 1 Show that is a solution to for . Solution We’ll need the first and second derivative to do this. Plug these as well as the function into the differential equation. So, does satisfy the differential equation and hence is a solution. Why then did I include the condition that ? I did not use this condition anywhere in the work showing that the function would satisfy the differential equation. To see why recall that In this form it is clear that we’ll need to avoid at the least as this would give division by zero. Also, there is a general rule of thumb that we’re going to run with in this class. This rule of thumb is : Start with real numbers, end with real numbers. In other words, if our differential equation only contains real numbers then we don’t want solutions that give complex numbers. So, in order to avoid complex numbers we will also need to avoid negative values of x.