Chapter 8: Q42P (page 430)

Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.
$y^{"} + 9 y = {\begin{cases} x, 0 < x < 1 \\ 0, - 1 < x < 0 \end{cases}$

Short Answer

Expert verified

Answer

The solution of function $y^{"} + 9 y = {\begin{cases} x, 0 < x < 1 \\ 0, - 1 < x < 0 \end{cases}$ is $a_{n} = \frac{1}{π} (1 - \sin x - \cos x)$ .

Step by step solution

Given information from question

The function is given as $y^{"} + 9 y = {\begin{cases} x, 0 < x < 1 \\ 0, - 1 < x < 0 \end{cases}$ .

The quadratic formula

The quadratic formula to find out the roots is,

$\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Expand f(x) in a Fourier series

The given function is

$y^{"} + 9 y = {\begin{cases} x, 0 < x < 1 \\ 0, - 1 < x < 0 \end{cases}$

Hereis a function of period

$f (x) = {\begin{cases} x, i f 0 < x < 1 \\ 0, i f - 1 < x < 0 \end{cases}$

The auxiliary equation is

$D^{2} + 9 = 0 D^{2} = - 9 D = \pm 3 i$

Calculate the complementary function

The complementary function is

$y_{c} = e^{3 x} (A \cos 3 x + B \sin 3 x) f (x) = \{\begin{cases} x i f < x < 1 \\ 0 i f - 1 < x < 0 \end{cases} a_{n} = \frac{1}{π} \int_{1}^{1} f (x) \cos n x d a_{n} = \frac{1}{π} \int_{1}^{0} 0 \cos n x d x + \int_{}^{1} x \cos n x d x$

Solve equation further

$a_{n} = \frac{1}{π} \int_{0}^{1} x \cos n x d x a_{n} = \frac{1}{π} {(x \cos n x + n x)}_{0}^{1} a_{n} = \frac{1}{π} (1 - \sin x - \cos x)$

Thus, the solution of functionis.

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Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.
$y^{"} + 9 y = {\begin{cases} x, 0 < x < 1 \\ 0, - 1 < x < 0 \end{cases}$

Short Answer

Step by step solution

Given information from question

The quadratic formula

Expand f(x) in a Fourier series

Calculate the complementary function

One App. One Place for Learning.

Most popular questions from this chapter

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Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.y"+9y={x,0<x<10,-1<x<0

Short Answer

Step by step solution

Given information from question

The quadratic formula

Expand f(x) in a Fourier series

Calculate the complementary function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Space Physics

Thermodynamics

Atoms and Radioactivity

Radiation

Electromagnetism

Study anywhere. Anytime. Across all devices.

Solve by use of Fourier series. Assume in each case that the right-hand side is a periodic function whose values are stated for one period.
$y^{"} + 9 y = {\begin{cases} x, 0 < x < 1 \\ 0, - 1 < x < 0 \end{cases}$