Question

Suppose \(p\) and \(q\) are continuous on \((a, b)\) and \(\left\\{y_{1}, y_{2}\right\\}\) is a fundamental set of solutions of $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$ on \((a, b)\). Show that if \(y\) is a solution of \((\mathrm{A})\) on \((a, b)\), there's exactly one way to choose \(c_{1}\) and \(c_{2}\) so that \(y=c_{1} y_{1}+c_{2} y_{2}\) on \((a, b)\).

Short answer

Question: Show that for any solution y of a given second-order linear homogeneous differential equation, there's exactly one way to choose the constants c_1 and c_2 such that y = c_1 y_1 + c_2 y_2, given that y_1 and y_2 are two linearly independent solutions of the equation. Answer: This can be shown by calculating the unique solutions for c_1 and c_2 using the Wronskian determinant, which is nonzero for linearly independent solutions y_1 and y_2. By deriving expressions for c_1 and c_2 in terms of the Wronskian determinant and substituting them back into the general solution y, it is demonstrated that there's exactly one way to choose c_1 and c_2 on the interval (a, b).

Step-by-step solution

Defining a second-order linear homogeneous differential equation

We are given a second-order linear homogeneous differential equation of the form: $$y'' + p(x)y' + q(x)y = 0$$ where \(p(x)\) and \(q(x)\) are continuous on \((a, b)\).

Assuming solutions

We are given two linearly independent solutions, denoted as \(y_1\) and \(y_2\). Also, we're given a general solution \(y\), which is a linear combination of the two fundamental solutions: $$y = c_1 y_1 + c_2 y_2$$ Our goal is to show that there's exactly one way to choose \(c_1\) and \(c_2\) to satisfy this equation on the interval \((a, b)\).

Differentiate the general solution

First, let's differentiate the general solution \(y\) with respect to \(x\): $$y' = c_1 y_1' + c_2 y_2'$$ Next, differentiate once more to obtain the second derivative: $$y'' = c_1 y_1'' + c_2 y_2''$$

Substitute the derivatives into the differential equation

Now, substitute the expressions for \(y\), \(y'\), and \(y''\) into the given second-order linear homogeneous differential equation to obtain: $$(c_1 y_1'' + c_2 y_2'') + p(x)(c_1 y_1' + c_2 y_2') + q(x)(c_1 y_1 + c_2 y_2) = 0$$

Define the Wronskian determinant

We will now calculate the Wronskian determinant \(W\) for the two fundamental solutions \(y_1\) and \(y_2\). Recall that the Wronskian determinant is given by: $$W = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'$$ Since \(y_1\) and \(y_2\) are linearly independent, their Wronskian determinant is nonzero.

Solve for \(c_1\) and \(c_2\)

Let's multiply both sides of the differential equation derived in Step 4 by the inverse of the Wronskian determinant to derive \(c_1\) and \(c_2\). The expression for \(c_1\) is given by: $$c_1 = \frac{1}{W} \begin{vmatrix} y & y_2 \\ y' & y_2' \end{vmatrix}$$ And the expression for \(c_2\) is given by: $$c_2 = \frac{1}{W} \begin{vmatrix} y_1 & y \\ y_1' & y' \end{vmatrix}$$

Conclusion

Since there is a unique solution for \(c_1\) and \(c_2\) in terms of the Wronskian determinant, there's exactly one way to choose the constants \(c_1\) and \(c_2\) such that \(y = c_1 y_1 + c_2 y_2\) on the interval \((a, b)\).

Key concepts

Fundamental Set of Solutions

In the realm of differential equations, a fundamental set of solutions is vital for understanding how different solutions can be related to form a general solution. When faced with a second-order linear homogeneous differential equation of the form \[ y'' + p(x)y' + q(x)y = 0 \]we often encounter two solutions, \(y_1(x)\) and \(y_2(x)\), that form what is called a \(\textit{fundamental set}\).

For these solutions, it's crucial they remain linearly independent. This independence means that one solution cannot be written as a constant multiple of the other. Why is this important? Because only linearly independent solutions are capable of spanning the solution space for the differential equation, which in layman terms means every possible solution of the differential equation can be expressed as a linear combination of these two solutions. This general solution can be expressed as:

• \(y(x) = c_1y_1(x) + c_2y_2(x)\)

where \(c_1\) and \(c_2\) are constants. This implies that any solution can be constructed by finding the right coefficients \(c_1\) and \(c_2\). This concept is fundamental because it provides a systematic way to write all solutions based on two basic solutions, hence maximizing the understanding of the differential equation's solution landscape.

Wronskian Determinant

The Wronskian determinant is a powerful tool used to determine whether a set of solutions are linearly independent. For a pair of functions \(y_1\) and \(y_2\) that are solutions to a differential equation, the Wronskian is calculated as:
\[ W(y_1, y_2) = \begin{vmatrix} y_1 & y_2 \ y_1' & y_2' \end{vmatrix} = y_1y_2' - y_2y_1' \]
If \(W(y_1, y_2) eq 0\), the functions \(y_1\) and \(y_2\) are linearly independent, guaranteeing that they form a fundamental set of solutions. This is crucial because a non-zero Wronskian ensures the solutions span the entire solution space of the differential equation.

Using the Wronskian has another practical benefit. When solving for the specific constants \(c_1\) and \(c_2\) in the general solution \(y = c_1y_1 + c_2y_2\), the Wronskian helps find these coefficients uniquely. With the expressions:

• \(c_1 = \frac{1}{W} \begin{vmatrix} y & y_2 \ y' & y_2' \end{vmatrix}\)
• \(c_2 = \frac{1}{W} \begin{vmatrix} y_1 & y \ y_1' & y' \end{vmatrix}\)

By dividing by the Wronskian, we ensure that these solutions are valid and unique, as long as \(W eq 0\). This provides a method to effectively solve the differential equation by determining the exact form of the solution based on known functions \(y_1\) and \(y_2\).

Second-Order Homogeneous Equations

Second-order homogeneous differential equations, such as\( y'' + p(x)y' + q(x)y = 0 \), are among the central types of differential equations studied in mathematics. The term "homogeneous" conveys that the equation is set to equal zero, which signifies it lacks a non-homogeneous term that could complicate the solution form.

The solutions of these equations are particularly interesting because they often depict natural systems' intrinsic behaviors, like mechanical systems without external forces. Therefore, understanding them can provide insights into motion, waves, and other phenomena.

When confronted with a second-order homogeneous differential equation:

• We start by finding two linearly independent solutions, referred to as a fundamental set.
• We utilize the Wronskian determinant to ensure these solutions are indeed independent.
• Finally, we express the general solution as a linear combination of these solutions.

What's pivotal is that these equations generally have solutions that are predictable and structured, leading to equations whose forms are exclusively based on the coefficients \(p\) and \(q\) being continuous. Consequently, they allow us to explore and develop methods that can solve even more complex systems by building from this basic understanding. Recognizing these characteristics influences how we approach broader, more intricate differential equations in our studies, offering a foundational knowledge that empowers deeper learning.