Problem 10 - Advanced Engineering Mathematics

Question

In Problems 9 and 10, find an interval centered about \(x=0\) for which the given initial-value problem has a unique solution. $$ y^{\prime \prime}+(\tan x) y=e^{x}, y(0)=1, y^{\prime}(0)=0 $$

Step-by-Step Solution

Answer

Unique solution in the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 1: Establish the Form of the Differential Equation

The given differential equation is \(y'' + \tan(x) y = e^x\). It is a second-order linear differential equation of the form \(y'' + p(x) y = g(x)\), where \(p(x) = \tan(x)\) and \(g(x) = e^x\).

Step 2: Identify the Condition for Unique Solutions

For the existence and uniqueness of solutions to be ensured, functions \(p(x) = \tan(x)\) and \(g(x) = e^x\) need to be continuous on an interval around \(x=0\).

Step 3: Determine Continuity of \( \tan(x) \)

The function \( \tan(x) \) is continuous wherever it is defined. It has discontinuities at points \( x = \frac{\pi}{2} + n\pi \) for integer \(n\). Around \(x=0\), \( \tan(x) \) is continuous.

Step 4: Establish an Interval for Continuity

Since the nearest discontinuity of \( \tan(x) \) from \(x=0\) is at \(x=\frac{\pi}{2}\), the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) is the largest interval centered around \(x=0\) where \( \tan(x) \) and \( e^x \) are continuous.

Step 5: Conclusion on Interval for Unique Solution

Therefore, the differential equation has a unique solution in the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\) centered about \(x=0\). This interval ensures the continuity of both \( \tan(x) \) and \(e^x\) in the solution process.

Key Concepts

Unique Solutions

In the world of differential equations, finding a unique solution to a problem is often crucial. When we say a solution is "unique," it means that under certain conditions, there is one and only one solution that satisfies the given initial conditions. This ensures that the problem has a mathematically consistent answer, and eliminates any ambiguity in solving it.

For our problem, the initial conditions are specified as:

• When \( x = 0 \), \( y = 1 \).
• The derivative at \( x = 0 \) is \( y'(0) = 0 \).

These initial conditions help us find a specific path or path curve among all possible solutions of the differential equation. The uniqueness of a solution guarantees that there is only one correct path that meets those initial conditions given the mathematical landscape defined by the equation.

Differential Equations

A differential equation is like a magic wand that talks about the relation between a function and its derivatives. In our exercise, the differential equation given is \( y'' + \tan(x) y = e^x \). This is a second-order linear differential equation:

• "Second-order" because it contains the second derivative \( y'' \).
• "Linear" because the dependent variable \( y \) and its derivatives appear to the power of one, without multiplication or division by each other.

Differential equations are essential in modeling real-world phenomena such as motion, heat, and electricity. They serve as the backbone of physics, engineering, and many other sciences. Understanding how to manipulate and solve them opens up a world of possibilities for describing how things change and behave dynamically.

One neat thing about linear differential equations is that they often lend themselves to solutions that can be represented with closed-form expressions, making them particularly practical for calculations.

Existence and Uniqueness Theorem

The Existence and Uniqueness Theorem is a powerhouse in the realm of differential equations. It assures us that under certain conditions, not only does a solution exist for a differential equation, but that solution is unique. This gives a concrete assurance that an equation behaves predictably under given initial conditions.

The theorem requires the involved functions to be continuous over the interval of concern. For example:

• \( p(x) = \tan(x) \) must be continuous; however, \( \tan(x) \) has discontinuities at \( x = \frac{\pi}{2} + n\pi \), which need careful attention.
• \( g(x) = e^x \) is well-behaved and continuous everywhere and causes no issues.

To find an interval centered around \( x = 0 \) where both functions are continuous, we consider the continuity of \( \tan(x) \), which breaks at \( \pm \frac{\pi}{2} \). Therefore, the largest interval around \( x = 0 \) where both functions are continuous is \((-\frac{\pi}{2}, \frac{\pi}{2})\).

This interval prominently showcases how the Existence and Uniqueness Theorem guides us in identifying the domain over which a unique solution can exist and be trusted across."