Q. 67 - Calculus

Question

The second-order differential equation  

$x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}+\left(x^2-p^2\right)=0$

where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_p\left(x\right)$. It may be shown that $J_p\left(x\right)$ is given by the following power series in x:  

$J_p\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+p)!2^{2k+p}}{x}^{2k+p}$

Graph the first four terms in the sequence of partial sums of $J_1\left(x\right)$. 

Step-by-Step Solution

Answer

The plot is 

Step 1.Given information

An expression is given as $J_p\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+p)!2^{2k+p}}{x}^{2k+p}$

Step 2. Plot of Four terms

The $J_1\left(x\right)$ is

$J_1\left(x\right)=\sum _{k=0}^{\infty }\frac{(-1{)}^k}{k!(k+1)!2^{2k+1}}{x}^{2k+1}$

Therefore the first four terms are 

$\frac{1}{2}x,\frac{1}{2}x-\frac{1}{16}{x}^3,\frac{1}{2}x-\frac{1}{16}{x}^3+\frac{1}{384}{x}^5,\frac{1}{2}x-\frac{1}{16}{x}^3+\frac{1}{384}{x}^5-\frac{1}{18432}{x}^7$

And the plot is