1. Consider the differential equation y\' ry where r is a constant. (Here is an
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1. Consider the differential equation y' ry where r is a constant. (Here is an example of equation with r 7: y' 7y.) Note that specifying y(0) uniquely determines the solution. Proving that result is non- trivial. Don't worry, I am not asking you to prove that here. But, FYI, it reduces to showing that if f' 0 over an interval, then f constant. This result is invoked in your undergraduate calculus courses, but usually only is proved in Advanced Calculus, or in an introduction to Analysis. Recall, from Calculus I, that by dividing both sides of the equation by y and then applying the substitution method (which is a symbolic technique for reversing the chain rule"), the general solution of equation can be obtained. It is of the form y aert where a is the initial value y(0) a. You should do the computations, if you don't remember the details. The differential equation can also be written in a more balanced way, y-ry30 Observe now that the set of all solutions to is a one-dimensional vector space, at least in the case when coefficients are real numbers. I am not asking you to include this result in your assignment, but it is a helpful warm-up to think about why this is true, and actually identify a basis here. For a basis, find a single function, multiples of which produce all solutions of e This is the equation that we will be really studying in the problem. One way that this is obtained is in Newtonian physics. By normalizing units, it is the equation for the motion of a unit mass on a horizontal spring with spring constant 1, along an idealized surface with friction zero. Note the similarity with the first-order equation Both equations )and (")represent a function that is proportional to one of its derivatives! And so, it is natural to guess that an appropriate exponential function might also solve equation on that hunch, tryExplanation / Answer
a>
y''+y= 0
This is a second order homogenous differential equation
the corresponding charactristic equation is :
r^2 + 1 = 0
r^2 = -1
r = +-sqrt(+-1)
or r1 = i and r2 = -i
now the general solution to such differential equation is given as :
y =e^(r1*t) + e^(r2*t)
=> y = e^(i*t) + r^(-i*t)
Hence r^(i*t) is a solution if r = +- i
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