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

Polar to cartesian conversion formulas, Polar to Cartesian Conversion Formu...

Polar to Cartesian Conversion Formulas x = r cos Θ y = r sin Θ Converting from Cartesian is more or less easy.  Let's first notice the subsequent. x 2 + y 2   = (r co

Prove that if x is a real number then [2x] = [x] + [x + ½ ], Prove that if...

Prove that if x is a real number then [2x] = [x] + [x + ½ ] Ans: Let us consider x be any real number. It comprises two parts: integer and fraction. With no loss of

Math, what is quantity ?

what is quantity ?

Continuous uniform distribution, Continuous Uniform Distribution Consid...

Continuous Uniform Distribution Consider the interest earned on a bank deposit. Let X equal the value after the decimal point. (Assume no rounding off to the nearest paise.) Fo

Economics, A mortgage lender seeks to maximize the expected value of its po...

A mortgage lender seeks to maximize the expected value of its portfolio. The portfolio, of course, is the sum of all of the mortgages in it, so no generality is lost by examining t

Subtangents & subnormals, show that the subtangent at any point on parabola...

show that the subtangent at any point on parabola y2 =4ax is twice the abscissa at that point.

Integers, hi i would like to ask you what is the answer for [-9]=[=5] grade...

hi i would like to ask you what is the answer for [-9]=[=5] grade 7

The width of a rectangle is 30.5% of its length, The width of a rectangle i...

The width of a rectangle is 30.5% of its length l. Write a formula for the area and perimeter of the rectangle in terms of l only

Explain simplifying rational expressions, Explain Simplifying Rational Expr...

Explain Simplifying Rational Expressions ? A rational expression, or algebraic fraction, is an expression in which you have a polynomial divided by a polynomial. Sometimes it

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