Complex root - fundamental set of solutions, Mathematics

Assignment Help:

Example: Back into the complex root section we complete the claim that

y1 (t ) = elt cos(µt)        and      y2(t) = elt sin(µt)

Those were a basic set of solutions.  Prove that they actually are.

Solution

Thus, to prove this we will require to take find the Wronskian for these two solutions and show that this isn't zero.

1105_Complex root - fundamental set of solutions.png

= elt cos(µt)( lelt sin(µt) + µ elt cos(µt)) - elt sin(µt)( lelt cos(µt) - µ elt sin(µt))

= µ e2lt cos2(µt) + µ e2lt sin2(µt)

= µ e2lt( cos2(µt) + sin2(µt))

= µ e2lt

Here, the exponential will never be zero and µ ≠ 0 whether it were we wouldn't have complex roots and so W ≠ 0. Thus, these two solutions are actually a fundamental set of solutions and hence the general solution in this case is. As:

 y (t ) = c1elt cos (mt ) + c2eltsin (mt)


Related Discussions:- Complex root - fundamental set of solutions

Computerised payroll package, How to calculate costs if you have a computer...

How to calculate costs if you have a computerised payroll package for your large business?

Graphs of sin x and cos x, Q. Graphs of Sin x and Cos x ? Ans. The...

Q. Graphs of Sin x and Cos x ? Ans. The sine and cosine functions are related to the path that an object might take around a circle. Suppose a dolphin was swimming over

Practice, #question.Mai is 3 years ypunger than twice the age of her brothe...

#question.Mai is 3 years ypunger than twice the age of her brother .If b represents .

Rules of logarithms, Rule 1 The logarithm of 1 to any base is 0. Pro...

Rule 1 The logarithm of 1 to any base is 0. Proof We know that any number raised to zero equals 1. That is, a 0 = 1, where "a" takes any value. Therefore, the loga

Find the probability density function, 1. The lifetime T (in days) of an el...

1. The lifetime T (in days) of an electrical component has reliability function given by: R(t) = e -0.01t for time t > 0. An electrical system consists of four such components. Th

Least common multiple (lcm), Before we look at this, let us learn wha...

Before we look at this, let us learn what a multiple is. Take any number say 3. Multiply this number with natural numbers. We obtain 3, 6, 9, 12, 15, 18,.........

Domain and range of a function , Domain and range of a functio:  One of th...

Domain and range of a functio:  One of the more significant ideas regarding functions is that of the domain and range of a function. In simplest world the domain of function is th

Levels of significance - rejection and acceptance regions, Levels of signif...

Levels of significance A level of significance is a probability value which is utilized when conducting tests of hypothesis. A level of significance is mostly the probability

Basic mathematics, I need help with my homework, I am to the edge right now...

I need help with my homework, I am to the edge right now with this w=5pq/2

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd