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

Interpretations of the derivative , Interpretations of the Derivative : ...

Interpretations of the Derivative : Before moving on to the section where we study how to calculate derivatives by ignoring the limits we were evaluating in the earlier secti

The length of the field is 2 more than twice the width field, Samantha owns...

Samantha owns a rectangular field that has an area of 3,280 square feet. The length of the field is 2 more than twice the width. What is the width of the field? Let w = the wid

Solving a quadratic equation, In polynomials you have seen expressi...

In polynomials you have seen expressions of the form x 2 + 3x - 4. Also we know that when an expression is equated to zero or some other expression, we cal

Determine the volume of the object, A rectangular container is 15 cm wide a...

A rectangular container is 15 cm wide and 5 cm long, and contains water to a depth of 8 cm. An object is placed in the water and the water rises 2.3 cm. Determine the volume of the

Integration by parts -integration techniques, Integration by Parts -Integra...

Integration by Parts -Integration Techniques Let's start off along with this section with a couple of integrals that we should previously be able to do to get us started. Fir

Solid mensuration, a circle is circumscribed about an equilateral triangle ...

a circle is circumscribed about an equilateral triangle whose side is 3 cm. find the area of the circle.

What is the measure of its width if its length is 3 inches, The perimeter o...

The perimeter of a rectangle is 21 inches. What is the measure of its width if its length is 3 inches greater than its width? Let x = the width of the rectangle. Let x + 3 = th

Probability, two coins are flipped once.what is the probability of getting ...

two coins are flipped once.what is the probability of getting two tails?

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