Problem 13 - Differential Equations and Linear Algebra

Question

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} \cosh 3 x+c_{2} \sinh 3 x, \quad y^{\prime \prime}-9 y=0$$.

Step-by-Step Solution

Answer

The function \(y(x) = c_1 \cosh 3x + c_2 \sinh 3x\) is a solution to the given differential equation, \(y'' - 9y = 0\), as we have verified by plugging in the second derivative. The maximum interval over which the solution is valid is the entire real number line, or \(\mathbb{R} = (-\infty, \infty)\).

Step 1: Calculate the first and second derivatives of y(x)

We are given the function, \(y(x) = c_1 \cosh 3x + c_2 \sinh 3x\). First, we need to find the derivatives of the function y(x): First, let's calculate the first derivative, \(y'(x)\): \[y'(x) = \frac{d}{dx}\left(c_1 \cosh 3x + c_2 \sinh 3x\right)\] Apply the chain rule: \[y'(x)=3c_1 \sinh 3x + 3c_2 \cosh 3x\] Next, calculate the second derivative, \(y''(x)\): \[y''(x) = \frac{d^2}{dx^2}\left(3c_1 \sinh 3x + 3c_2 \cosh 3x\right)\] Apply the chain rule again: \[y''(x)= 9c_1 \cosh 3x + 9c_2 \sinh 3x\]

Step 2: Verify if the second derivative satisfies the given differential equation

Now, we need to check whether the second derivative satisfies the given differential equation, \(y^{\prime \prime} - 9y = 0\). Plug the function \(y(x)\) and its second derivative \(y''(x)\) into the differential equation: \[9c_1 \cosh 3x + 9c_2 \sinh 3x - 9(c_1 \cosh 3x + c_2 \sinh 3x) = 0\] Factor out 9 from both terms, \[9(c_1 \cosh 3x + c_2 \sinh 3x) = 9y\] Since the equation holds true, we have verified that the given function y(x) is a solution to the differential equation.

Step 3: Determine the maximum interval over which the solution is valid

Since both \(\cosh x\) and \(\sinh x\) are defined for all real numbers, the solution is valid for any value of x on the entire real number line. Hence, the maximum interval over which the solution is valid is: \[\mathbb{R} = (-\infty, \infty)\]

Key Concepts

General Solution

In the realm of differential equations, a general solution encompasses all possible solutions, primarily characterized by the presence of arbitrary constants. For the differential equation \(y'' - 9y = 0\), the described function \(y(x) = c_1 \cosh 3x + c_2 \sinh 3x\) represents such a solution. The coefficients \(c_1\) and \(c_2\) are arbitrary constants, which means we can generate infinite solutions by varying their values.

Typically, solving a second-order linear differential equation like this one involves finding two linearly independent solutions and combining them. In our context:

• \(\cosh 3x\) and \(\sinh 3x\) are the independent solutions
• Coefficients \(c_1\) and \(c_2\) modulate how much each function influences the general solution

This approach is fundamental for understanding how solutions behave under varying conditions and initial values.

Hyperbolic Functions

Hyperbolic functions, \(\cosh\) and \(\sinh\), play a crucial role in solving differential equations due to their unique properties. Much like their trigonometric counterparts, they are defined as:

• \( \cosh x = \frac{e^x + e^{-x}}{2} \)
• \( \sinh x = \frac{e^x - e^{-x}}{2} \)

These functions are essential for describing scenarios that involve hyperbolas, such as certain physics problems.

In solving the equation \(y'' - 9y = 0\), hyperbolic functions naturally emerge as solutions. Unlike the trigonometric functions, hyperbolic functions are defined for all real numbers, mirroring growth and decay behaviors. This characteristic allows them to be immensely flexible and applicable:

• The derivative of \(\cosh x\) is \(\sinh x\)
• The derivative of \(\sinh x\) is \(\cosh x\)

Thus, these functions are incredibly advantageous in both analytic and practical contexts.

Verification of Solutions

When working with differential equations, verifying a solution is a key step. To determine if \(y(x) = c_1 \cosh 3x + c_2 \sinh 3x\) is a solution to \(y'' - 9y = 0\), we follow these steps:

• Compute the first and second derivatives of \(y(x)\)
• Substitute these derivatives back into the original differential equation
• Check if the left-hand side equals the right-hand side (in this case, zero)

By calculating the derivatives:

• First derivative: \(y'(x) = 3c_1 \sinh 3x + 3c_2 \cosh 3x\)
• Second derivative: \(y''(x) = 9c_1 \cosh 3x + 9c_2 \sinh 3x \)

Substituting gives us \(y'' - 9y = 0\), showing the left-hand side simplifies to zero, confirming that our function meets the equation's requirements.

Lastly, we consider the interval of validity. Hyperbolic functions have no restrictions on \(x\), so the solution is valid over all real numbers, or \((-\infty, \infty)\). This step ensures the solution's completeness and appropriateness for the given equation across the desired domain.