Q35E Use the result of Problem 34 to ... [FREE SOLUTION]

91影视

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: Q35E (page 327)

Use the result of Problem 34 to construct a third-order differential equation for which ${x, 鈥� sinx, 鈥� cosx}$ is a fundamental solution set.

Short Answer

Expert verified

Thus, the required differential equation is $xy''' + xy' - y'' - y = 0 .$

Step by step solution

Use the given fundamental solution set,

Given the fundamental solution set,

${x, 鈥� sinx, 鈥� cosx}$

We have,

$f_{1} = x, 鈥� f_{2} = sinx, 鈥� f_{3} = cosx$

Given the equation of problem 34 is,

$| \begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) & y \\ f_{1}' (x) & f_{2}' (x) & f_{3}' (x) & y' \\ f_{1}'' (x) & f_{2}'' (x) & f_{3}'' (x) & y'' \\ f_{1}''' (x) & f_{2}''' (x) & f_{3}'' (x) & y''' \end{matrix} | = 0 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� ....... (1)$

Now Use the given equation of Problem 34,

Substitute $f_{1} = x, 鈥� f_{2} = sinx, 鈥� f_{3} = cosx$ in the equation (1),

$| \begin{matrix} x & sinx & cosx & y \\ 1 & cosx & - sinx & y' \\ 0 & - sinx & - cosx & y'' \\ 0 & - cosx & sinx & y''' \end{matrix} | = 0$

Now find the determinate in the above equation,

$\begin{matrix} x | \begin{matrix} cosx & - sinx & y' \\ - sinx & - cosx & y'' \\ - cosx & sinx & y''' \end{matrix} | - 1 | \begin{matrix} sinx & cosx & y \\ - sinx & - cosx & y'' \\ - cosx & sinx & y''' \end{matrix} | = 0 \\ x (- y''' - y') + y'' + y = 0 \\ - xy''' - xy' + y'' + y = 0 \\ xy''' + xy' - y'' - y = 0 \end{matrix}$

Thus, the required differential equation is $xy''' + xy' - y'' - y = 0 .$

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91影视

Use the result of Problem 34 to construct a third-order differential equation for which ${x, 鈥� sinx, 鈥� cosx}$ is a fundamental solution set.

Short Answer

Step by step solution

Use the given fundamental solution set,

Now Use the given equation of Problem 34,

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91影视

Use the result of Problem 34 to construct a third-order differential equation for which{x,鈥�sinx,鈥�cosx} is a fundamental solution set.

Short Answer

Step by step solution

Use the given fundamental solution set,

Now Use the given equation of Problem 34,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use the result of Problem 34 to construct a third-order differential equation for which ${x, 鈥� sinx, 鈥� cosx}$ is a fundamental solution set.