MAT 275 Online Activity 3 A differential equation is called autonomous when it does not explicitly depend on the independent va riable. For example, 푦 ² = 푦 is autonomous but 푦 ² = 푦 + í‘¡ is not. If the independent variable is time, then we can explain what autonomous really means in terms of application: an autonomous equation does not know what tim e it is . Take for example the equation 푦 ² = 푘 푦 with a positive constan t k. I t describes exponential growth , say, of a bacteria culture. The equation says that the growth rate of the culture is proportional to how many bacteria there are already . The more bacteria there are, the faster the culture grows. It would not make any sense to have an explicit t dependence in that equation, because bacteria don ‘ t know what time it is. A 19 th century and a 21 st century scientist cultivating the same bacteria under the same condition would obser ve the exact same growth curves relative to their own starting times of the experiment. Differential equations that express n atural laws with time being the independent variable ar e always autonomous, because to the best of our current knowledge, th e laws of nat ure do not change over time. The same experiment, conducted yesterday, today or tomorrow, must always produce the same outcome relative to the starting time. This leads us directly to the following fact : if 푦 ( í‘¡ ) is a solution of an autonomous differential equation, then any time – shifted version of that function , i.e. 푦 ( í‘¡ ˆ’ í‘ ) , must also be a solution. It must be , because the equation does not know what time it is. If it allows for a behavior at one time, it must allow for the same behavior at all times. 1. The first goal of this activity is to demonstrate t h at this hyp othesis is true for first order equations. Let us start by assuming that the given first order autonomous equation is 푦 ² = í‘“ ( 푦 ) . Demonstrate, using the rules of differentiation, that if 푦 1 ( í‘¡ ) is a solution of that equation, then 푦 2 ( í‘¡ ) = 푦 1 ( í‘¡ ˆ’ í‘ ) , where c is a constant, is a solution as well. You must show all algebraic steps. 2. The fact that shifted solutions of autonomous equations are again solutions has a n important consequence for the gen eral solution of such equations: the general solution of an autonomous equation is shift – invariant. If you introduce a time shift into the general solution it ‘ s still the general solution. For example, we know that the general solution of 푦 ² = 푦 is 푦 = í¶ í‘’ í‘¡ . By the theorem we just learned, then , 푦 = í¶ í‘’ í‘¡ ˆ’ 1 is also the general solution . Use algebra to demonstrate this. Show that every function of the form í¶ í‘’ í‘¡ ˆ’ 1 is also a function of the form í¶ í‘’ í‘¡ and vice versa . Do not just write equations. You must explain your reasoni ng. 3. S hift – invariance of the general solution can lead to dramatic computational simpli fications when you have to satisfy initial conditions. Let us consider the equation 푦 ² ² + 2 푦 ² + 10 푦 = 0 . It s general solution is 푦 ( í‘¡ ) = í‘’ ˆ’ í‘¡ ( í´ cos 3 í‘¡ + íµ sin 3 í‘¡ ) . Say we want to find the particular solution that satisfies the conditions 푦 ( 1 ) = 0 and 푦 ² ( 1 ) = 1 . D irect substitution into the general solution leads to a system of equations for A and B that is very complicated. Demonstrate how, by exploiting shift – invariance of the general solution, you can f ind the desired particular solution very easily. For a custom paper on the above topic, place your order now! What We Offer: ¢ On-time delivery guarantee ¢ PhD-level writers ¢ Automatic plagiarism check ¢ 100% money-back guarantee ¢ 100% Privacy and Confidentiality ¢ High Quality custom-written papers
