Question

Bessel’s Equation

In Problems 1-6 use (1) to find the general solution of the given differential equation on $\left(0,\infty \right)$.

5. $\mathbf{xy}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{xy}\mathbf{=}\mathbf{0}$

Short answer

The general solution of the given differential equation is

$\mathrm{y}={\mathrm{C}}_1{\mathrm{J}}_0\left(\mathrm{X}\right)+{\mathrm{C}}_2{\mathrm{Y}}_0\left(\mathrm{X}\right)$

Step-by-step solution

Bessel’s equation of order v

The special form of equation (1) encountered by Bessel eventually carried out a systematic study of the properties of the solutions of the general equation.

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

We have the form in

$xy\text{'}\text{'}+y^{\text{'}}+xy=0$

multiplying by (x) on both sides,

$$\begin{gathered} \left(xy\text{'}\text{'}+y^{\text{'}}+xy\right)\left(x\right)=0\left(x\right) \\ \Rightarrow x^{2}y\text{'}\text{'}+x{y}^{\text{'}}+\left(x^2-0\right)y=0 \end{gathered}$$

Use Forbinous methodto solve this differential equation

$y=\sum _{n=0}^{\infty }{c}_n{x}^{n+r}$

Step 2: Find the first and second derivative

Substituting the given differential equation ends up with two roots,

r₁ = v and r₂ = - v

Identify the solution to have the form of Bessel function of First kind of order v.

The two solutions are:

$$\begin{gathered} y_1\left(x\right)=J_v\left(x\right) \\ {J}_V\left(X\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!\Gamma \left(1+v+n\right)}{\left(\frac{x}{2}\right)}^{2n+v} \\ {y}_2\left(x\right)=J_{-v}\left(x\right) \\ {J}_{-V}\left(X\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!\Gamma \left(1-v+n\right)}{\left(\frac{x}{2}\right)}^{2n-v} \end{gathered}$$

which are linearly independent for v = non-integer

The two solutions are not linearly independent and one of the solution is constant multiple of the other, for v = integer.

Using Bessel function of Second kind,obtain another solution linearly independent with J_v(X) ,

$$\begin{gathered} y_3\left(x\right)=Y_v\left(x\right) \\ {Y}_v\left(x\right)=\frac{\cos v\pi J_v\left(x\right)-J_{-v}\left(x\right)}{\sin v\pi } \end{gathered}$$

For the given differential equation, the root is

v² = 0

v = 0

Therefore, we have a single solution, which is

$y_1\left(x\right)=J_0\left(x\right)$

We can obtain another solution, which is linearly independent with y₁, using Bessel function of Second kind.

The second solution is,

y₂ (x) = y₀(x)

By using the Superposition Principle,the general solution is

$y=C_1{J}_0\left(X\right)+C_2{Y}_0\left(X\right)$

Where C₁ and C₂ are arbitrary constants

Therefore, the general solution of the given differential equation $xy+y^{\text{'}}+xy=0on\left(0,\infty \right)isy=C_1{J}_0\left(X\right)+C_2{Y}_0\left(X\right)$ on is