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Q8.3-31E

Expert-verified
Found in: Page 444

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# (a) Use the first formula in (30) and Problem 32 to find the spherical Bessel functions $${j_1}(x)$$ and $${j_2}(x)$$.(b) Use a graphing utility to plot the graphs of $${j_1}(x)$$ and $${j_2}(x)$$ in the same coordinate plane.

(a) The values are $${j_1}(x) = \frac{1}{{{x^2}}}sinx - \frac{1}{x}cosx$$ and $${j_2}(x) = \left( {\frac{3}{{{x^3}}} - \frac{1}{x}} \right)sinx - \frac{3}{{{x^2}}}cosx$$.

(b) The graph has been plotted.

See the step by step solution

## Step 1:Define Spherical Bessel’s equation.

Bessel functions of half-integral order areused to dene two more important functions:

\begin{aligned}{j_n}(x) = \sqrt {\frac{\pi }{{2x}}} {J_{n + 1/2}}(x)\\{y_n}(x) = \sqrt {\frac{\pi }{{2x}}} {Y_{n + 1/2}}(x)\end{aligned}

The function $${j_n}(x)$$ is called the spherical Bessel function of the first kind and $${y_n}(x)$$is the spherical Bessel function of the second kind.

## Step 2: Find the value of $${j_1}(x)$$.

Using the above formula and the result of Problem 32. Let,

\begin{aligned}{j_1}(x) &= \sqrt {\frac{\pi }{{2x}}} {J_{3/2}}\\ &= \sqrt {\frac{\pi }{{2x}}} \left( {\sqrt {\frac{2}{{\pi {x^3}}}} sinx - \sqrt {\frac{2}{{\pi x}}} cosx} \right)\\ &= \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi {x^3}}}} sinx - \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi x}}} cosx\\ &= \frac{1}{{{x^2}}}sinx - \frac{1}{x}cosx\end{aligned}

## Step 3: Find the value of $${j_2}(x)$$.

Let,

\begin{aligned}{j_2}(x) &= \sqrt {\frac{\pi }{{2x}}} {J_{5/2}}\\ &= \sqrt {\frac{\pi }{{2x}}} \left( {\left( {\sqrt {\frac{{18}}{{\pi {x^5}}}} - \sqrt {\frac{2}{{\pi x}}} } \right)sinx - \sqrt {\frac{{18}}{{\pi {x^3}}}} cosx} \right)\\ &= \left( {\sqrt {\frac{\pi }{{2x}} \cdot \frac{{18}}{{\pi {x^5}}}} - \sqrt {\frac{\pi }{{2x}} \cdot \frac{2}{{\pi x}}} } \right)sinx - \sqrt {\frac{\pi }{{2x}} \cdot \frac{{18}}{{\pi {x^3}}}} cosx\\ &= \left( {\sqrt {\frac{9}{{{x^6}}}} - \sqrt {\frac{1}{{{x^2}}}} } \right)sinx - \sqrt {\frac{9}{{{x^4}}}} cosx\\ &= \left( {\frac{3}{{{x^3}}} - \frac{1}{x}} \right)sinx - \frac{3}{{{x^2}}}cosx\end{aligned}

## Step 4: Graph of the Bessel’s function.

Let the graph of $${j_1}(x)$$ and $${j_2}(x)$$ be,