Chapter 6: Q35E (page 327)

Use the result of Problem 34 to construct a third-order differential equation for which ${x, sinx, cosx}$ 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,

${x, sinx, cosx}$

We have,

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

Given the equation of problem 34 is,

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

Now Use the given equation of Problem 34,

Substitute $f_{1} = x, f_{2} = sinx, f_{3} = cosx$ 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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Use the result of Problem 34 to construct a third-order differential equation for which ${x, sinx, cosx}$ 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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Use the result of Problem 34 to construct a third-order differential equation for which{x, sinx, 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

Decision Maths

Applied Mathematics

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

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