Question

. The differential equation for transverse vibrations of a string whose density increases linearly from one end to the other is \(y^{\prime \prime}+(A x+B) y=0\), where \(A\) and \(B\) are constants. Find the general solution of this equation in terms of Bessel functions. Hint: Make the change of variable \(A x+B=A u\).

Short answer

The general solution is \( y(x) = c_1 J_0(\sqrt{A} (x + \frac{B}{A})) + c_2 Y_0(\sqrt{A} (x + \frac{B}{A})) \).

Step-by-step solution

Change of Variable

Introduce a new variable given by the hint. Set \(u = \frac{Ax+B}{A}\), which implies \(Ax + B = Au\). The differential equation becomes \(y'' + (Au)y=0\).

Simplifying the Differential Equation

Rewrite the transformed differential equation: \(y'' + Auy = 0\).

Standard Form of Bessel's Equation

Recognize that the transformed equation is a form of the Bessel differential equation, which is\(u^2 y'' + u y' + (u^2 - u^2)y = 0\) for \(u =0\).

General Solution in Terms of Bessel Functions

The general solution of the given differential equation, in terms of Bessel functions, is \(y(u) = c_1 J_0(\sqrt{A} u) + c_2 Y_0(\sqrt{A} u)\), where \(J_0\) and \(Y_0\) are Bessel functions of the first and second kind, respectively.

Substitute Back the Original Variable

Replace \(ueq \frac{Ax+B}{A}\) back into the solution to get the general solution in terms of the original variable x:\[y(x) = c_1 J_0\left(\sqrt{A} \frac{Ax+B}{A}\right) + c_2 Y_0\left(\sqrt{A} \frac{Ax+B}{A}\right)\]. Simplifying the argument of Bessel functions:\[y(x) = c_1 J_0\left(\sqrt{A} (x + \frac{B}{A})\right) + c_2 Y_0\left(\sqrt{A} (x + \frac{B}{A})\right)\].

Key concepts

Transverse String Vibrations

Transverse string vibrations refer to the oscillations that occur perpendicular to the length of a string. These oscillations are caused by external forces or initial displacements.

When the density of the string varies, the equation governing the vibrations changes accordingly. In this case, we are dealing with a string whose density increases linearly from one end to the other.
This gives rise to a differential equation with variable coefficients. The problem can be modeled by the equation: \(y'' + (Ax + B)y = 0\), where the coefficients depend on the position along the string.

Variable Change in ODE

To simplify the differential equation given, we perform a change of variables. This rearrangement helps in converting the equation into a more familiar form.

Given the hint in the problem, we introduce a new variable: \(u = \frac{Ax + B}{A}\). By this substitution, the new equation becomes: \(y'' + (Au)y = 0\). Changing variables is a common technique in solving differential equations, as it can transform a complex equation into a more manageable one.

Bessel Differential Equation

The simplified differential equation \(y'' + Auy = 0\) can be recognized as a form of the Bessel differential equation. The standard form of a Bessel differential equation is: \[u^2 y'' + u y' + (u^2 - v^2)y = 0\],
where \(v\) is a constant called the order of the Bessel function. Bessel functions, \(J_v(u)\) and \(Y_v(u)\), solve this equation and are known as Bessel functions of the first and second kinds, respectively.

In this problem, the differential equation takes a specific form that corresponds to: \[u^2 y'' + uy' + u^2 y = 0\],
which is solved by Bessel functions with order zero. So, the general solution involves \(J_0(\text{{argument}})\) and \(Y_0(\text{{argument}})\).

Density Variation

The variation in density across the string affects the form of the differential equation that models the transverse vibrations. Here, the density increases linearly with the position along the string, leading to variable coefficients in the equation.

The differential equation \(y'' + (Ax + B)y = 0\) accounts for this linear variation. By solving this equation through a change of variables and recognizing it as a Bessel differential equation, we can express the solution in terms of Bessel functions.
The final general solution to the problem is expressed as: \[y(x) = c_1 J_0\big(\text{{scaled variable}}\big) + c_2 Y_0\big(\text{{scaled variable}}\big)\]
where the argument is derived from the original variables and constants, reflecting the linearly increasing density along the string.