Chapter 4: Q6E (page 172)

The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 4 y' + 7 y = 0$

Short Answer

Expert verified

The auxiliary equation for the given differential equation $y'' - 4 y' + 7 y = 0$ has complex roots and its general solution is $y (t) = e^{2 t} (c_{1} \cos (\sqrt{3} t) + c_{2} \sin (\sqrt{3} t))$ .

Step by step solution

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $α \pm i β$ , then the general solution is given as:

$y (t) = c_{1} e^{α t} c o s β t + c_{2} e^{α t} s i n β t$ .

Finding the roots of the auxiliary equation.

Given differential equation is $y'' - 4 y' + 7 y = 0 .$

Then the auxiliary equation is $r^{2} - 4 r + 7 = 0 .$

$r = \frac{4 \pm \sqrt{4^{2} - 4 \times 1 \times 7}}{2 \times 1} r = \frac{4 \pm \sqrt{16 - 28}}{} r = \frac{4 \pm 2 i \sqrt{3}}{} r = 2 \pm \sqrt{3} i$

Final answer.

Therefore, the general solution is:

$y (t) = e^{2 \times t} (c_{1} c o s (\sqrt{3} t) + c_{2} s i n (\sqrt{3} t)) y (t) = e^{2 t} (c_{1} \cos (\sqrt{3} t) + c_{2} \sin (\sqrt{3} t))$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 4 y' + 7 y = 0$

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Applied Mathematics

Calculus

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

The auxiliary equation for the given differential equation has complex roots. Find a general solution.y''-4y'+7y=0

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Applied Mathematics

Calculus

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 4 y' + 7 y = 0$