Q4.3-8E The auxiliary equation for the g... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q4.3-8E (page 172)

The auxiliary equation for the given differential equation has complex roots. Find a general solution $4 y'' - 4 y' + 26 y = 0$ .

Short Answer

Expert verified

The auxiliary equation for the given differential equation $4 y'' - 4 y' + 26 y = 0$ has complex roots and its general solution isrole="math" localid="1654065502147" $y (t) = e^{\frac{1}{2} t} (c_{1} \cos (\frac{5 t}{2}) + c_{2} \sin (\frac{5 t}{2}))$ .

Step by step solution

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $伪卤 i 尾$ , then the general solution is given as:

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

Finding the roots of the auxiliary equation.

Given differential equation is $4 y'' - 4 y' + 26 y = 0 .$

Then the auxiliary equation is $4 r^{2} - 4 r + 26 = 0$

The roots of the auxiliary equation are:

role="math" localid="1654065783488" $r = \frac{4 卤 \sqrt{4^{2} - 4 脳 4 脳 26}}{2 脳 4} r = \frac{4 卤 \sqrt{16 - 416}}{8} r = \frac{4 卤 20 i}{8} r = \frac{1}{2} 卤 \frac{5 i}{2}$

Final answer.

Therefore, the general solution is:

$y (t) = e^{\frac{}{} 脳 t} (c_{1} \cos (\frac{5 t}{2}) + c_{2} \sin (\frac{5 t}{2})) y (t) = e^{\frac{}{} t} (c_{1} \cos (\frac{5 t}{2}) + c_{2} \sin (\frac{5 t}{2}))$

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91影视

The auxiliary equation for the given differential equation has complex roots. Find a general solution $4 y'' - 4 y' + 26 y = 0$ .

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation.

Final answer.

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91影视

The auxiliary equation for the given differential equation has complex roots. Find a general solution 4y''-4y'+26y=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

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

The auxiliary equation for the given differential equation has complex roots. Find a general solution $4 y'' - 4 y' + 26 y = 0$ .