By Florian Cajori
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Additional resources for An introduction to the theory of equations
The solution resembles eít as time goes backward and e3t as time goes forward. 1 44 Another example of a first-order linear ODE is Newton’s law of cooling. Suppose you have a cup of coffee whose temperature at the moment is 170°. Suppose also that the ambient temperature is 70°. Can you find a function y(t) that gives the temperature of your coffee as a function of t? Newton’s law of cooling says that the rate of cooling is directly proportional to the difference between the current temperature and the ambient temperature.
Lowering the step size further and further usually gives better and better approximations. 2 55 Perhaps the easiest way nowadays to invoke Euler’s method is to use a spreadsheet. 3. 1 and displayed the corresponding sequence of straight line segments. Unfortunately, no numerical algorithm is flawless; we can always find a differential equation that breaks a given numerical method. For example, consider the differential equation yƍ = etsin(y).
To solve these equations, we use the “guess and check” method. We know how to solve the homogeneous part of this equation; the solution is y(t) = Cekt. We then make an appropriate guess to find the general solution of the nonhomogeneous equation. Lecture 5: Methods for ¿nding Explicit Solutions As an example of this method, consider yƍ + y = e3t. The solution of the homogeneous equation yƍ + y = 0 is y(t) = Ceít. So we would make a guess of the form y(t) = Ceít + Ae3t. The question is, what is A?
An introduction to the theory of equations by Florian Cajori