Question

Bessel’s Equation

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

2.

localid="1664866362361" ${\mathbf{x}}^{\mathbf{2}}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}{\mathbf{xy}}^{\mathbf{\text{'}}}\mathbf{+}\left(x^2-1\right)\mathbf{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution of the given differential equation is

$y\left(x\right)=C_1{J}_1\left(X\right)+C_2{Y}_1\left(X\right)$

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

$x^{2}y\text{'}\text{'}+x{y}^{\text{'}}+\left(x^2-1\right)y=0$

Use Forbinous methodto solve this differential equation

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

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:

$J_V\left(X\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!\Gamma \left(1+v+n\right)}{\left(\frac{x}{2}\right)}^{2n+v}$

$J_{-V}\left(X\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!\Gamma \left(1-v+n\right)}{\left(\frac{x}{2}\right)}^{2n-v}$

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)

$Y_v\left(x\right)=\frac{\cos \pi v{J}_v\left(x\right)-J_{-v}\left(x\right)}{\sin \pi v}$

The two roots are:

v²= 1

v = 1 and v = -1

Substituting the values of v:

The two linearly independent solutions are,

$y_1\left(x\right)=J_1\left(x\right)$

$y_2\left(x\right)=Y_1\left(x\right)$

By using theSuperposition Principle, $y=C_1{J}_{1/3}\left(X\right)+C_2{J}_{-1/3}\left(X\right)$

$y\left(x\right)=C_1{J}_1\left(X\right)+C_2{Y}_1\left(X\right)$

Where C₁ and C₂ are arbitrary constants

Therefore, the general solution of the given differential equation

$x^{2}y\text{'}\text{'}+x{y}^{\text{'}}+\left(x^2-1\right)y=0$ on$\left(0,\infty \right)$is$y\left(x\right)=C_1{J}_1\left(X\right)+C_2{Y}_1\left(X\right)$