Q25E 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 4: Q25E (page 172)

Solve the given initial value problem. $y'' - 2 y' + 2 y = 0;$ $y (蟺) = e^{蟺},$ $y' (蟺) = 0$

Short Answer

Expert verified

The solution of the given initial value $y'' - 2 y' + 2 y = 0$ is $y (t) = e^{t} (- cost + sint)$ when $y (蟺) = e^{蟺}$ and $y' (蟺) = 0$ .

Step by step solution

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $伪卤颈尾$ , then the general solution is given as: $y (t) = c_{1} e^{伪迟} 肠辞蝉尾迟 + c_{2} e^{伪迟} 蝉颈苍尾迟$

Finding the roots of the auxiliary equation

Given differential equation is $y'' - 2 y' + 2 y = 0 .$

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

$\begin{array}{l} r = \frac{2 卤 \sqrt{2^{2} - 4 脳 1 脳 2}}{2 脳 1} \\ r = \frac{2 卤 \sqrt{4 - 8}}{} \\ r = \frac{2 卤 \sqrt{- 4}}{} \\ r = \frac{2 卤 2 i}{} \\ r = 1 卤 i \end{array}$

Therefore, the general solution is:

$\begin{matrix} y (t) = e^{1 脳 t} (c_{1} \cos (t) + c_{2} \sin (t)) \\ = e^{t} (c_{1} \cos (t) + c_{2} \sin (t)) \end{matrix}$

Finding the values of c1 and c2

Given initial conditions are $y (蟺) = e^{蟺}$ and $y' (蟺) = 0$

$\begin{array}{l} y (蟺) = e^{蟺} (c_{1} \cos (蟺) + c_{2} \sin (蟺)) \\ - c_{1} e^{蟺} = e^{蟺} \\ c_{1} = - 1 \end{array}$

And $y' (t) = e^{t} (c_{1} cost + c_{2} sint) + e^{t} (- c_{1} sint + c_{2} cost)$

Then $y' (蟺) = e^{蟺} (c_{1} \cos (蟺) + c_{2} \sin (蟺)) + e^{蟺} (- c_{1} \sin (蟺) + c_{2} \cos (蟺))$

$\begin{matrix} - e^{蟺} (c_{1} + c_{2}) = 0 \\ c_{1} + c_{2} = 0 \\ c_{2} = - c_{1} \end{matrix}$

Then $c_{2} = 1$

Therefore, the solution is $y (t) = e^{t} (- cost + sint)$

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91影视

Solve the given initial value problem. $y'' - 2 y' + 2 y = 0;$ $y (蟺) = e^{蟺},$ $y' (蟺) = 0$

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation

Finding the values of c1 and c2

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91影视

Solve the given initial value problem. y''-2y'+2y=0;y(蟺)=e蟺,y'(蟺)=0

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation

Finding the values of c1 and c2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Mechanics Maths

Statistics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

Solve the given initial value problem. $y'' - 2 y' + 2 y = 0;$ $y (蟺) = e^{蟺},$ $y' (蟺) = 0$