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Hence we have a linked system of first order linear differential equations. The expressions of linear equations are linear combinations of functions.Įach equation in our system is a linear combination of functions x( t) and y( t) and both have one derivative of either x or y.A linked system of differential equations has at least one function or derivative that is found in at least two equations in the system.A n th order differential equation only involves up to the n th derivative of any function.A differential equation is an equation that contains both function(s) and their derivatives.
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Introduction to Systems of Linear Differential Equations Nevertheless, our model is useful for analyzing the general outline of the Cold War arms race and predicting its outcomes. Note: We must use our model with a degree of skepticism because there are far more factors affecting the arms race than just the two countries' defense spending, such as available money/resources and the spending requirements of other important programs. Equation y′( t) states that the Soviet Union lowers its budget by d for every dollar it spent the previous year, and raises it by c dollars for every dollar in the United States's budget. In other words, equation x′( t) states that the United States lowers its budget by a dollars for every dollar it spent the previous year, and raises it by b dollars for every dollar in the Soviet Union's budget. This is a linked system of first order linear differential equations. Let x 0 and y 0 be the initial( t = 0) defense spending of the United States and the Soviet Union respectively The following equations fit these conditions. The model's starting point is the year in which the arms race started, when t = 0. Correspondingly, the rate of Soviet defense spending depends negatively on its current defense spending and positively on the United States's current defense spending. defense spending depends negatively on its current defense spending and positively on the Soviet Union's current defense spending. Likewise, y′( t) is the derivative of y over t and it represents how much the Soviet spending changes over t years. Then x′( t) is the derivative of x over t and it represents how much the U.S. Let x( t) and y( t) be the defense spending of the United States and the Soviet Union respectively at time t (in years). Since derivatives are the mathematical tool to analyze change, we will build our model around derivatives. With this dynamic, we can analyze the defense spending of each country by analyzing the change in defense spending from year to year. However, the more the Soviet Union spends on defense, the more likely the United States will be to increase its defense spending to not be left behind. The more the United States is already spending on defense, the less willing it is to spend on defense the following year. These fluctuating changes are connected with each power's analysis of both its own defense spending and that of the other power. From that point on, the spending of the two powers pushed and pulled, staying within a fairly narrow range (the two higher points of United States defense spending around 19 are related to the Korean and Vietnam Wars). The United States responded in kind beginning in 1948 (Figure 1). Note that, starting in 1947, the Soviet Union rapidly increased its defense spending (Figure 2). What type of model would fit best? Consider these two images. We will attempt to build a simple model to see the effects of the nuclear arms race. During this period, both powers invested tremendous resources into their technology and weaponry, worried that the other was pulling ahead. capitalism), caused a political and psychological war during the second half of the 20th century, now known as the Cold War. The competition for nuclear might, along with the countries' different ideologies (communism vs.
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This marked the beginning of a long and expensive arms race between the two powers. The Soviet Union, while an ally of the United States during WWII, feared the bomb and spent the next few years developing their own atomic bomb, finally detonating their first nuclear weapon in 1949.
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In 1945, the United States dropped atomic bombs on the Japanese cities Hiroshima and Nagasaki, ending World War II and establishing itself as a new superpower. 2.4 Stability of the Two Dimensional System.2.1.2 Back to the Two Dimensional System.1.1 Introduction to Systems of Linear Differential Equations.