Problem 38
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
Verified Answer
\( \cosh x \) and \( \sinh x \) are solutions as they are linear combinations of \( e^x \) and \( e^{-x} \).
1Step 1: Express hyperbolic functions in terms of exponential functions
The hyperbolic cosine and sine functions are defined as combinations of exponential functions. Specifically, \( \cosh x = \frac{e^x + e^{-x}}{2} \) and \( \sinh x = \frac{e^x - e^{-x}}{2} \). This means they are directly related to the solutions \( y_1 = e^x \) and \( y_2 = e^{-x} \).
2Step 2: Verify linearity of the differential equation
A homogeneous linear differential equation has the property that any linear combination of its solutions is also a solution to the equation. Since \( y_1 = e^x \) and \( y_2 = e^{-x} \) are solutions, any linear combination \( ay_1 + by_2 \) (where \( a \) and \( b \) are constants) is also a solution.
3Step 3: Express \( y_3 \) and \( y_4 \) as linear combinations
\( y_3 = \cosh x = \frac{1}{2}(e^x + e^{-x}) \) is a linear combination of \( y_1 \) and \( y_2 \) with \( a = \frac{1}{2} \), \( b = \frac{1}{2} \). Similarly, \( y_4 = \sinh x = \frac{1}{2}(e^x - e^{-x}) \) is a linear combination with \( a = \frac{1}{2} \), \( b = -\frac{1}{2} \).
4Step 4: Conclude hyperbolic functions are solutions
Since both \( y_3 = \cosh x \) and \( y_4 = \sinh x \) can be written as linear combinations of the solutions \( y_1 \) and \( y_2 \), they are also solutions to the homogeneous linear differential equation due to the principle of superposition.
Key Concepts
Hyperbolic FunctionsExponential FunctionsPrinciple of Superposition
Hyperbolic Functions
Hyperbolic functions, namely hyperbolic sine (\( \sinh \)) and hyperbolic cosine (\( \cosh \)), play a key role in mathematics, similar to trigonometric functions. However, instead of angles on a circle, they relate to the geometry of a hyperbola.
The two primary hyperbolic functions are defined this way:
- The hyperbolic cosine is given by \( \cosh x = \frac{e^x + e^{-x}}{2} \).- The hyperbolic sine is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \).
These definitions use exponential functions, linking them to the solutions of certain differential equations.
Hyperbolic functions are handy in solving differential equations because they satisfy similar relations as sine and cosine, like the identity \( \cosh^2 x - \sinh^2 x = 1 \). Their unique properties make them useful in handling problems involving hyperbolic geometry and other advanced mathematical constructs.
The two primary hyperbolic functions are defined this way:
- The hyperbolic cosine is given by \( \cosh x = \frac{e^x + e^{-x}}{2} \).- The hyperbolic sine is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \).
These definitions use exponential functions, linking them to the solutions of certain differential equations.
Hyperbolic functions are handy in solving differential equations because they satisfy similar relations as sine and cosine, like the identity \( \cosh^2 x - \sinh^2 x = 1 \). Their unique properties make them useful in handling problems involving hyperbolic geometry and other advanced mathematical constructs.
Exponential Functions
Exponential functions are mathematical expressions that describe growth or decay. In the form \( e^x \), they feature prominently in finance, biology, and, not least, in differential equations.
In this context, exponential functions like \( e^x \) and \( e^{-x} \) are solutions to specific types of homogeneous linear differential equations. These equations are characterized by their reliance on exponential functions to describe their solutions.
- The general solutions often involve combinations of exponential solutions.- These combinations can take different forms depending on the equation's parameters.
Understanding exponential functions and their behavior is crucial. They help find the underlying nature of solutions for diverse differential equations, including those that give rise to hyperbolic functions.
In this context, exponential functions like \( e^x \) and \( e^{-x} \) are solutions to specific types of homogeneous linear differential equations. These equations are characterized by their reliance on exponential functions to describe their solutions.
- The general solutions often involve combinations of exponential solutions.- These combinations can take different forms depending on the equation's parameters.
Understanding exponential functions and their behavior is crucial. They help find the underlying nature of solutions for diverse differential equations, including those that give rise to hyperbolic functions.
Principle of Superposition
The principle of superposition is a foundational concept in linear algebra and differential equations. It states that if you have multiple solutions to a homogeneous linear differential equation, any linear combination of those solutions will also be a solution.
This principle stems from the linearity of the equation itself. If \( y_1 \) and \( y_2 \) are solutions to the equation, then for any constants \( a \) and \( b \), the function \( ay_1 + by_2 \) is also a solution.
In the context of hyperbolic and exponential functions, this principle allows us to express hyperbolic functions, such as \( \cosh x \) and \( \sinh x \), in terms of exponential functions \( e^x \) and \( e^{-x} \). This linear combination forms the basis for demonstrating that these functions solve the same homogeneous differential equations.
Understanding superposition can provide significant insights when analyzing physical systems and mathematical models that follow linear behaviors.
This principle stems from the linearity of the equation itself. If \( y_1 \) and \( y_2 \) are solutions to the equation, then for any constants \( a \) and \( b \), the function \( ay_1 + by_2 \) is also a solution.
In the context of hyperbolic and exponential functions, this principle allows us to express hyperbolic functions, such as \( \cosh x \) and \( \sinh x \), in terms of exponential functions \( e^x \) and \( e^{-x} \). This linear combination forms the basis for demonstrating that these functions solve the same homogeneous differential equations.
Understanding superposition can provide significant insights when analyzing physical systems and mathematical models that follow linear behaviors.
Other exercises in this chapter
Problem 38
$$ \text { solve the given boundary-value problem. } $$ $$ \begin{array}{llll} y^{\prime \prime}+4 y & 0, y(0) & 0, y(\pi) & 0 \end{array} $$
View solution Problem 38
Solve the given initial-value problem. $$ \frac{d^{2} x}{d t^{2}}+9 x=5 \sin 3 t, x(0)=2, x^{\prime}(0)=0 $$
View solution Problem 38
In Problems 33-38, solve the given differential equation subject to the indicated conditions. $$ 2 y^{\prime \prime}=3 y^{2}, y(0)=1, y^{\prime}(0)=1 $$
View solution Problem 38
In Problems 37 and 38, solve the given initial-value problem. $$ \frac{d^{2} x}{d t^{2}}+9 x=5 \sin 3 t, x(0)=2, x^{\prime}(0)=0 $$
View solution