Q21E Solve the given initial value pr... [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 6: Q21E (page 332)

Solve the given initial value problem
_{$\begin{array}{l} y''' 鈭� 4 y'' + 7 y' 鈭� 6 y = 0 \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \end{array}$}

Short Answer

Expert verified

_{The solution is}_{$y (t) = e^{2 t} 鈭� \sqrt{2} e^{t} \sin \sqrt{2} t$}

Step by step solution

Basic differentiation

The Sum rule says the derivative of a sum of functions is the sum of their derivatives. The Difference rule says the derivative of a difference of functions is the difference of their derivatives.

Solving by basic differentiation:

We will do the following question on the basis of basic differentiation

$\begin{array}{l} r^{3} 鈭� 4 r^{2} + 7 r 鈭� 6 = 0 \\ r^{3} 鈭� 4 r^{2} + 7 r 鈭� 6 = (r 鈭� 2) (r^{2} 鈭� 2 r + 3) = 0 \\ y (t) = c_{1} e^{2 t} + c_{2} e^{t} \sin \sqrt{2} t + c_{3} e^{t} \cos \sqrt{2} t \\ y' (t) = 2 c_{1} e^{2 t} + c_{2} e^{t} \sin \sqrt{2} t + \sqrt{2} c_{2} e^{t} \cos \sqrt{2} t + c_{3} e^{t} \cos \sqrt{2} t 鈭� \sqrt{2} c_{3} e^{t} \sin \sqrt{2} t \\ y'' (t) = 4 c_{1} e^{2 t} + c_{2} e^{t} \sin \sqrt{2} t + \sqrt{2} c_{2} e^{t} \cos \sqrt{2} t + \sqrt{2} c_{2} e^{t} \cos \sqrt{2} t 鈭� \sqrt{2} c_{2} e^{t} \sin \sqrt{2} t + c_{3} e^{t} \cos \sqrt{2} t 鈭� \sqrt{2} c_{3} e^{t} \sin \sqrt{2} t 鈭� \sqrt{2} c_{3} e^{t} \sin \sqrt{2} t 鈭� \sqrt{2} c_{3} e^{t} \cos \sqrt{2} t \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \\ y (t) = e^{2 t} 鈭� \sqrt{2} e^{t} \sin \sqrt{2} t \end{array}$

Hence, the final answer is:

$y (t) = e^{2 t} 鈭� \sqrt{2} e^{t} \sin \sqrt{2} t$

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91影视

Solve the given initial value problem
_{$\begin{array}{l} y''' 鈭� 4 y'' + 7 y' 鈭� 6 y = 0 \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \end{array}$}

Short Answer

Step by step solution

Basic differentiation

Solving by basic differentiation:

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91影视

Solve the given initial value problem y'''鈭�4y''+7y'鈭�6y=0y(0)=1y'(0)=0y''(0)=0

Short Answer

Step by step solution

Basic differentiation

Solving by basic differentiation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Geometry

Calculus

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve the given initial value problem
_{$\begin{array}{l} y''' 鈭� 4 y'' + 7 y' 鈭� 6 y = 0 \\ y (0) = 1 \\ y' (0) = 0 \\ y'' (0) = 0 \end{array}$}