91Ó°ÊÓ

Let the Bessel equation be x²y"+xy'+(x²-n²)y=0. This equation has two linearly independent solutions for a fixed value of n . A Bessel equation of the first kind, indicated by J_n(x), is one of these solutions that may be derived usingFrobinous approach.

y₁ = x^aJ_p(bx^c)

y₁ = x^aJ_-p(bx^c)

At ,x = 0 this solution is regular. The second solution, which is singular at,x = 0 , is represented by Y_n(x) and is called a Bessel function of the second kind.

$y_3=x^a\left[\frac{\left(cosp\mathrm{Ï€}\right)J_p\left(b{x}^c\right)-J_{-p}\left(b{x}^c\right)}{sinp\mathrm{Ï€}}\right]$

Let the given be 4x"e^-0.1tx=0,x(0)=1,x'(0)=$\frac{-1}{2}$.That has the general solution x(t)=c₁J₀$\left(\frac{a}{2}\sqrt{\frac{k}{m}{e}^{-at/2}}\right)$+ C₂Y₀$\left(\frac{2}{a}\sqrt{\frac{k}{m}{e}^{-at/2}}\right)$, where C₁ and C₂ aare arbitrary constants

The given DE has the form of mx"+ke^-atx=0 . That yields, m= 4, k =1, a=0.1

Hence, the general solution becomes,

x(t)= C₁J0role="math" localid="1663946251828" $\left(\frac{2}{0.1}\sqrt{\frac{1}{4}{e}^{-0.1t/2}}\right)$+C₂Y₉$\left(\frac{2}{0.1}\sqrt{\frac{1}{4}{e}^{-0.1t/2}}\right)$

=C₁J₀(10e^-t/20)+C₂Y₀(10e^-t/20)

Apply the initial conditions.

x(0)=C₁J₀(10e⁰)+C₂Y₀(10e⁰)

1=C₁J₀(10)+C₂Y₀(10) … (1)

x'(t)=$\frac{d}{dx}\left[C_1{J}_0\left(10e^{-t/20}\right)\right]$+$\frac{d}{dx}\left[C_2{Y}_0\left(10e^{-t/20}\right)\right]$

=

… (2)