Problem 38 - A First Course in Differential Equations with Modeling Applications

Question

Suppose that \(y_{1}=e^{x}\) and \(y_{2}=e^{-x}\) are two solutions of a homogeneous linear differential equation. Explain why \(y_{3}=\cosh x\) and \(y_{4}=\sinh x\) are also solutions of the equation.

Step-by-Step Solution

Answer

\( \cosh x \) and \( \sinh x \) are linear combinations of \( e^x \) and \( e^{-x} \), hence they are solutions too.

Step 1: Confirm Given Solutions

We have two functions, \( y_1 = e^x \) and \( y_2 = e^{-x} \), that are solutions to a homogeneous linear differential equation. This implies that the equation is of the form \( y'' + py' + qy = 0 \) where \( p \) and \( q \) are constants.

Step 2: Express Hyperbolic Functions

Recall the definitions of hyperbolic functions: \( \cosh x = \frac{e^x + e^{-x}}{2} \) and \( \sinh x = \frac{e^x - e^{-x}}{2} \). These are linear combinations of \( e^x \) and \( e^{-x} \).

Step 3: Use Linearity of Differential Equations

A homogeneous linear differential equation with constant coefficients is linear, i.e., if \( y_1 \) and \( y_2 \) are solutions, then any linear combination \( ay_1 + by_2 \) is also a solution. As both \( \cosh x \) and \( \sinh x \) are linear combinations of \( e^x \) and \( e^{-x} \), they are solutions as well.

Step 4: Conclusion

Since \( \cosh x \) and \( \sinh x \) can be expressed as linear combinations of \( e^x \) and \( e^{-x} \), which are given solutions, they are thus solutions to the same homogeneous linear differential equation due to the linearity property.

Key Concepts

Solutions of Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In other words, they describe how a particular quantity changes over time or space. When we resolve these equations, we find functions that satisfy the equation. These solutions are crucial as they help us model real-life phenomena, such as physics and engineering problems. To solve a differential equation, there are various methods available. The choice of method depends heavily on the type of differential equation we are dealing with. For a homogeneous linear differential equation of the form \[ y'' + py' + qy = 0 \]it implies that the right side of the equation is zero. This form leads to the notion that the overall nature of the change in the system is consistent or "homogeneous." If functions like \(e^x\) and \(e^{-x}\) solve the differential equation, any other function expressed as a linear combination of these—a function constructed using these solutions—will also solve the same equation. Thus, hyperbolic functions, which are combinations of these exponential functions, fit the solution set of such differential equations.

Hyperbolic Functions

Hyperbolic functions are analogs of regular trigonometric functions. However, instead of circles, they relate to hyperbolas. The two basic hyperbolic functions are the hyperbolic cosine and hyperbolic sine, defined as:

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

These definitions tell us that \(\cosh x\) and \(\sinh x\) are built using exponential functions. They resemble trigonometric identities but have unique properties suitable for hyperbolic geometry.Since hyperbolic functions are combinations of exponential functions, they are especially useful in solving differential equations where exponential functions appear as solutions. They often simplify the algebraic structure of solutions by grouping exponential terms neatly.In the context of homogeneous linear differential equations, these functions reveal their versatility: given that \(e^x\) and \(e^{-x}\) are solutions to a given equation, both \(\cosh x\) and \(\sinh x\) become natural candidates for solutions due to their composition.

Linearity of Differential Equations

The linearity of a differential equation is a property that significantly simplifies finding solutions. This property states that if two functions \(y_1\) and \(y_2\) are solutions of a linear differential equation, then any linear combination of these solutions is also a solution. In mathematical terms, for a homogeneous linear differential equation, this means:\[ ay_1 + by_2 \text{ is also a solution for any constants } a \text{ and } b \]This principle is incredibly powerful because it allows us to generate a whole family of solutions once we know just a couple of solutions. In our scenario, since \(y_1 = e^x\) and \(y_2 = e^{-x}\) solve the equation, any function composed as \[ a e^x + b e^{-x} \] is also a solution. Given the hyperbolic identities, \(\cosh x\) and \(\sinh x\) naturally conform to this linearity principle, as they are straightforward combinations of these exponentials. This linearity is crucial since it tells us that solutions can intermix, forming even more complex functions that still fit within the same equation framework.