Problem 34 - The Calculus with Analytic Geometry

Question

The partial differential equation for a vibrating string is $$ \frac{\partial^{2} u}{\partial t^{2}}=a^{2} \frac{\partial^{2} u}{\partial x^{2}} $$ Show that if \(f\) is a function of \(x\) satisfying the equation \(d^{2} f / d x^{2}+\lambda^{2} f(x)=0\) and \(g\) is a function of \(t\) satisfying the equation \(d^{2} g / d t^{2}+a^{2} \lambda^{2} g(t)=0\), then if \(u=f(x) g(t)\), the partial differential equation is satisfied. \(a\) and \(\lambda\) are constants.

Step-by-Step Solution

Answer

The given function \(u = f(x)g(t)\) satisfies the PDE as both sides simplify correctly showing the separation of variables method works.

Step 1: Express the solution

Let the solution be of the form \(u = f(x)g(t)\), where \(f\) is a function of \(x\) and \(g\) is a function of \(t\).

Step 2: Compute the second partial derivative with respect to time

Calculate the second partial derivative of \(u\) with respect to \(t\): \[ \frac{\text{d}^2 u}{\text{d} t^2} = f(x) \frac{\text{d}^2 g(t)}{\text{d} t^2}. \]

Step 3: Compute the second partial derivative with respect to space

Calculate the second partial derivative of \(u\) with respect to \(x\): \[ \frac{\text{d}^2 u}{\text{d} x^2} = g(t) \frac{\text{d}^2 f(x)}{\text{d} x^2}. \]

Step 4: Substitution into the PDE

Substitute the derived expressions back into the given PDE \[ \frac{\text{d}^2 u}{\text{d} t^2} = a^2 \frac{\text{d}^2 u}{\text{d} x^2}. \] This yields: \[ f(x) \frac{\text{d}^2 g(t)}{\text{d} t^2} = a^2 g(t) \frac{\text{d}^2 f(x)}{\text{d} x^2}. \]

Step 5: Separate variables

Separate the equation using the given \(f(x)\) and \(g(t)\) functions. Dividing both sides by \(f(x)g(t)\), we get: \[ \frac{1}{g(t)} \frac{\text{d}^2 g(t)}{\text{d} t^2} = a^2 \frac{1}{f(x)} \frac{\text{d}^2 f(x)}{\text{d} x^2}. \]

Step 6: Apply the given conditions

Apply the conditions \(\frac{\text{d}^2 f(x)}{\text{d} x^2} + \lambda^2 f(x) = 0\) and \(\frac{\text{d}^2 g(t)}{\text{d} t^2} + a^2 \lambda^2 g(t) = 0\) to rewrite the separated equation: \[ \frac{-a^2 \lambda^2 g(t)}{g(t)} = a^2 \frac{-\lambda^2 f(x)}{f(x)}. \]

Step 7: Simplify and conclude

On simplifying, both sides equal \(-a^2 \lambda^2\). Thus, the original partial differential equation is satisfied with the given conditions. Hence, it is validated that \(u = f(x)g(t)\) is a solution for the PDE.

Key Concepts

Second Partial Derivative

In understanding the vibrating string problem, knowing how to compute second partial derivatives is crucial. A second partial derivative is essentially the derivative of the first derivative, representing how a function changes as two variables independently change.

For instance, if we have the function \( u = f(x)g(t) \), the second partial derivative of \( u \) with respect to time \( t \) is given by:
\(\frac{\text{d}^2 u}{\text{d} t^2} = f(x) \frac{\text{d}^2 g(t)}{\text{d} t^2}\).

Similarly, the second partial derivative of \( u \) with respect to space \( x \) is:
\(\frac{\text{d}^2 u}{\text{d} x^2} = g(t) \frac{\text{d}^2 f(x)}{\text{d} x^2}\).

These calculations involve taking the derivative of the function \(u\) twice with respect to each variable. It's important to follow through each step carefully, ensuring that we correctly apply the rules of differentiation.

Understanding these derivatives is the foundation for solving the Partial Differential Equation (PDE) since they help in transforming the PDE into a more manageable equation.

Separation of Variables

Separation of Variables is a technique used to solve partial differential equations, including those involving vibrating strings.

The idea is to represent the solution \( u \) as a product of two functions, each depending on different variables, say \( x \) and \( t \):\( u = f(x)g(t)\). This method simplifies the PDE by allowing us to split it into two ordinary differential equations.

For example, substituting \( u = f(x)g(t) \) into the PDE \(\frac{\text{d}^2 u}{\text{d} t^2} = a^2 \frac{\text{d}^2 u}{\text{d} x^2} \) we get:
\(\frac{1}{g(t)}\frac{\text{d}^2 g(t)}{\text{d} t^2} = a^2 \frac{1}{f(x)}\frac{\text{d}^2 f(x)}{\text{d} x^2}\).

This equation can now be separated into two different equations because the left side depends solely on \( t \) and the right side solely on \( x \). Importantly, for the equality to hold for all \( x \) and \( t \), both sides must equal the same constant.

By setting them equal to \(-\text{constant} \), typically referred to as \( \lambda^2 \), we create two ordinary differential equations:

• \(\frac{\text{d}^2 f(x)}{\text{d} x^2} + \lambda^2 f(x) = 0\)
• \(\frac{\text{d}^2 g(t)}{\text{d} t^2} + a^2 \lambda^2 g(t) = 0\)

Boundary Conditions

Boundary conditions are additional constraints that solutions to PDEs must satisfy. For the vibrating string problem, these conditions specify the behavior of the string at its ends (boundaries).

For instance, if the ends of the string are fixed, we might have conditions like:
\( u(0, t) = 0 \) and \( u(L, t) = 0 \), where \( L \) is the length of the string.

These conditions ensure that the solutions not only satisfy the differential equation but also match the physical constraints of the problem.

For our separated functions, \( f(x) \) and \( g(t) \), the boundary conditions help determine the specific forms of these functions. For instance:

• The boundary conditions for \( f(x) \) might lead to a solution in the form of sine or cosine functions with specific wavelengths.
• Timing conditions for \( g(t) \) might define how the string vibrates over time, often leading to sinusoidal solutions as well.

Applying these conditions brings us closer to the final particular solution matching the physical scenario being studied.