Question

In this problem we outline a different derivation of Euler's formula. (a) Show that \(y_{1}(t)=\cos t\) and \(y_{2}(t)=\sin t\) are a fundamental set of solutions of \(y^{\prime \prime}+\) \(y=0 ;\) that is, show that they are solutions and that their Wronskian is not zero. (b) Show (formally) that \(y=e^{i t}\) is also a solution of \(y^{\prime \prime}+y=0 .\) Therefore, $$ e^{i t}=c_{1} \cos t+c_{2} \sin t $$ for some constants \(c_{1}\) and \(c_{2}\). Why is this so? (c) Set \(t=0\) in Eq. (i) to show that \(c_{1}=1\) (d) Assuming that Eq. ( 14) is true, differentiate \(E q\). (i) and then set \(t=0\) to conclude that \(c_{2}=i .\) Use the values of \(c_{1}\) and \(c_{2}\) in Eq. ( i ) to arrive at Euler's formula.

Short answer

Question: Using a different method, derive Euler's formula, which states that \(e^{it} = \cos t + i \sin t\). Answer: Euler's formula can be derived through the following steps: (a) Verify that \(\cos t\) and \(\sin t\) are solutions to the differential equation \(y'' + y = 0\) and calculate their Wronskian. (b) Show that \(e^{it}\) is also a solution to the differential equation and express it as a linear combination of the two previous solutions. (c) Find the constant \(c_1\) by plugging \(t=0\) into the equation. (d) Differentiate the equation, plug \(t=0\) to find \(c_2\), and substitute the values of \(c_1\) and \(c_2\) to obtain Euler's formula. Following these steps, we can derive Euler's formula: \(e^{it} = \cos t + i \sin t\).

Step-by-step solution

(a) Verify Solutions and Calculate Wronskian

To verify that \(\cos t\) and \(\sin t\) are solutions of the differential equation, we have to check if substituting these functions into the equation makes it true. For \(\cos t\), we have: \(y_1 = \cos t\) \(y_1' = -\sin t\) \(y_1'' = -\cos t\) Now, substitute into the equation: \(y_1'' + y_1 = (-\cos t) + (\cos t) = 0\) For \(\sin t\), we have: \(y_2 = \sin t\) \(y_2' = \cos t\) \(y_2'' = -\sin t\) Now, substitute into the equation: \(y_2'' + y_2 = (-\sin t) + (\sin t) = 0\) Thus, both \(\cos t\) and \(\sin t\) are solutions to the given differential equation. To calculate the Wronskian \(W\) of \(\cos t\) and \(\sin t\), we have: \(W = \begin{vmatrix} \cos t & \sin t\\ -\sin t & \cos t\\ \end{vmatrix} = \cos^2 t + (\sin t)^2 = 1\) Since the Wronskian is not zero, \(\cos t\) and \(\sin t\) form a fundamental set of solutions.

(b) Verify \(e^{it}\) as a Solution and Express as Linear Combination

To show that \(e^{it}\) is a solution, we need to check if it satisfies the differential equation. We find its derivatives: \(y = e^{it}\) \(y' = ie^{it}\) \(y'' = -e^{it}\) Now, substitute into the equation: \(y'' + y = (-e^{it}) + (e^{it}) = 0\) So, \(e^{it}\) is also a solution. According to the theorem of linear combination of solutions, if \(\cos t\) and \(\sin t\) form a fundamental set of solutions, then any other solution can be expressed as a linear combination: \(e^{it} = c_1 \cos t + c_2 \sin t\)

(c) Find \(c_1\) by setting \(t=0\)

We plug in \(t=0\) into the equation: \(e^{i(0)} = c_1 \cos(0) + c_2 \sin(0)\) \(1 = c_1(1) + (0)\) \(c_1 = 1\)

(d) Find \(c_2\), Plug Constant Values, and Get Euler's Formula

Differentiate the equation with respect to \(t\): \(\frac{d}{dt}(e^{it}) = \frac{d}{dt}(c_1 \cos t + c_2 \sin t)\) \(ie^{it} = -c_1 \sin t + c_2 \cos t\) Now, set \(t=0\) in the differentiated equation: \(ie^{i(0)} = -c_1 \sin(0) + c_2 \cos(0)\) \(i = c_2(1)\) \(c_2 = i\) Use the values of \(c_1\) and \(c_2\) in the equation from part (b): \(e^{it} = 1 \cdot \cos t + i \sin t\) Thus, we have derived Euler's Formula: \(e^{it} = \cos t + i \sin t\), using a different approach.

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In the context of the exercise, we encounter a second-order linear homogeneous differential equation with constant coefficients: \( y'' + y = 0 \). Such equations are foundational in physics and engineering as they often describe wave motions and vibrations.

To solve this type of equation, it is crucial to find functions whose second derivative (acceleration or curvature) negates the original function, implying a kind of repetitive or oscillatory motion. The functions \(\cos(t)\) and \(\sin(t)\), which represent fundamental wave forms, are classic solutions to this equation. Not only do they oscillate, but the rate at which they oscillate exactly offsets their position, resulting in the equation being satisfied.

Wronskian

The Wronskian is a determinant used in the theory of differential equations to determine whether a set of solutions is linearly independent. This concept is crucial when dealing with homogeneous differential equations.

In our exercise, after verifying that \(\cos(t)\) and \(\sin(t)\) are solutions to the differential equation, we compute their Wronskian. The value of the Wronskian, \( W = 1 \), confirms that the two functions are linearly independent and thus, form a fundamental set of solutions. This means that any solution to the differential equation can be expressed as a linear combination of these two functions. Without this critical step, we wouldn't be able to confirm the uniqueness of the solutions.

Complex Exponentials

Complex exponentials are expressions of the form \( e^{it} \), where \( i \) is the imaginary unit and \( t \) is a real number. This concept is deeply related to Euler's Formula, which links exponential functions to trigonometric functions using complex numbers.

In the problem, we demonstrate that \( e^{it} \) is a solution to the differential equation. This is remarkable because it bridges the seemingly unconnected exponential and trigonometric functions. By doing so, complex exponentials provide a powerful tool for solving differential equations and form the basis for Fourier analysis, which has extensive applications in signal processing and other fields.

Linear Combinations of Solutions

The principle of superposition allows us to form new solutions of linear homogeneous differential equations by taking linear combinations of existing solutions. In our example, we are told that \( e^{it} \) is a solution to the differential equation, and we must express it as a linear combination of the two fundamental solutions \(\cos(t)\) and \(\sin(t)\).

We determine the coefficients of this combination by evaluating the equation at \( t = 0 \) which leads to \( c_1 = 1 \), and taking the derivative and evaluating at \( t = 0 \) to find \( c_2 = i \). These steps are crucial for proving Euler's Formula, as they mathematically validate that the complex exponential function is indeed a harmonic combination of two perpendicular oscillatory motions represented by \(\cos(t)\) and \(\sin(t)\).