Question

Find the solution of the given initial value problem. Plot the graph of the solution and describe how the solution behaves as \(x \rightarrow 0\). \(4 x^{2} y^{\prime \prime}+8 x y^{\prime}+17 y=0, \quad y(1)=2, \quad y^{\prime}(1)=-3\)

Short answer

In summary, we have analyzed the given second-order homogeneous linear differential equation and attempted to obtain its solution using the variation of parameters method. However, due to the inability to define the functions \(y_1(x)\) and \(y_2(x)\) using elementary functions, we could not provide an explicit expression for the general solution and its graph. Additionally, the behavior of the solution as \(x \rightarrow 0\) is arbitrary, as all coefficients vanish in this limit.

Step-by-step solution

Analyze the given problem

We are given a second-order homogeneous linear differential equation: $$4x^2y'' + 8xy' + 17y = 0$$ with initial conditions \(y(1) = 2\) and \(y'(1) = -3\). We must find the solution of this equation, analyze its behavior as \(x\rightarrow 0\) and also plot its graph.

Find the complementary function

To obtain the complementary function, let us write the given differential equation as: $$y'' + \frac{8x}{4x^2}y' + \frac{17}{4x^2}y = 0$$ Now, compare it with the general form of the second-order ordinary differential equation: $$y'' + p(x)y' + q(x)y = 0$$ For our problem, \(p(x) = \frac{8}{4x}\) and \(q(x) = \frac{17}{4x^2}\). Now, we find the integrating factor, which is given by: $$\rho(x) = e^{\int p(x) dx} = e^{\int \frac{2}{x} dx}$$ $$\rho(x) = e^{2\ln x}= x^2$$ Let's multiply the given equation by the integrating factor, we get: $$x^2y'' + 2x^3y' + 17xy = 0$$

Solve the equation with variation of parameters

The general solution to the given equation can be written as \(y_p(x) = c_1y_1(x) + c_2y_2(x)\). Using the variation of parameters method, let \(y(x) = u_1(x)y_1(x) + u_2(x)y_2(x)\). Substitute this into the transformed equation and use the Wronskian method, it turns out that the easiest way to determine the particular solution for our problem is by setting: \(u_1'(x)y_1(x) + u_2'(x)y_2(x) = 0\) and \(u_1'(x)y_1'(x) + u_2'(x)y_2'(x) = \frac{-17}{x^2}\). Solve these equations to find the functions \(u_1(x)\) and \(u_2(x)\), then compute: $$y_p(x) = y_1(x)u_1(x) + y_2(x)u_2(x)$$ Here, we must emphasize that finding expressions for \(y_1(x)\) and \(y_2(x)\) is not possible using elementary functions, which means that further steps involving these functions are not practical.

Analyze the behavior of the solution and plot the graph

By applying the initial conditions, we can determine the constants \(c_1\) and \(c_2\). $$y_p(1) = c_1y_1(1) + c_2y_2(1) = 2$$ $$y_p'(1) = c_1y_1'(1) + c_2y_2'(1) = -3$$ However, as stated before, without expressions for \(y_1(x)\) and \(y_2(x)\), solving these equations is not possible using elementary functions. Therefore, we cannot provide an explicit expression for the final solution and its graph. We can recognize that the terms \(x^2y''\), \(xy'\), and \(y\) are balanced for this equation near the point \(x=0\). This is because as \(x\rightarrow 0\), the powers of \(x\) in these terms become closer to zero, resulting in terms that approach zero. Since all coefficients vanish as \(x\rightarrow 0\), the behavior of the solution becomes arbitrary in this limit, and a full analysis of the solution is not possible without an explicit expression for the general solution.

Key concepts

Second-Order Homogeneous Linear Differential Equation

A second-order homogeneous linear differential equation is a type of differential equation that involves derivatives up to the second order. The general form is given by: \[ y'' + p(x)y' + q(x)y = 0 \] where \( y'' \) represents the second derivative of the function \( y \), \( y' \) is the first derivative, and \( y \) is the function itself. Here, \( p(x) \) and \( q(x) \) are functions of \( x \). These equations are called homogeneous because the right-hand side is zero, indicating no external forces or inputs are acting upon the system.

In the given problem, we have \[ 4x^2y'' + 8xy' + 17y = 0 \] This equation represents such a system, and it is our task to find a function \( y \) that satisfies the equation for all \( x \) within its domain. Solving these equations sometimes involves converting them into their standard form to simplify analyzing and solving the problem.

Variation of Parameters

The Variation of Parameters is a method used to find particular solutions of non-homogeneous linear differential equations. While we usually use it for non-homogeneous equations, in some cases, like this problem involving homogeneous equations, it helps in constructing solutions where direct methods aren't viable.

The idea is to express the solution as a combination of two parts: the complementary function (coming from the homogeneous equation) and a particular solution that accounts for non-homogeneity or, in complex cases, variations in the coefficients. For the equation,\[ y(x) = u_1(x)y_1(x) + u_2(x)y_2(x) \]where \( y_1 \) and \( y_2 \) are solutions to the associated homogeneous equation, and \( u_1 \) and \( u_2 \) are functions found by solving a system of equations derived from the original equation.

Variation of Parameters allows flexibility when specific solutions cannot be found using usual methods, especially when the equation coefficients vary with \( x \).

Complementary Function

Finding the complementary function is crucial when solving differential equations, as it represents the part of the solution derived purely from the homogeneous equation. For second-order linear homogeneous equations, it involves finding two linearly independent solutions, \( y_1(x) \) and \( y_2(x) \). These functions form the basis of the complementary solution given by:\[ y_c(x) = c_1y_1(x) + c_2y_2(x) \]

The complementary function comprises these solutions, where \( c_1 \) and \( c_2 \) are constants determined by initial or boundary conditions. In the exercise, we transform the original equation through simplification and manipulation to find the complementary function.

Although direct solutions \( y_1(x) \) and \( y_2(x) \) may not be expressible using elementary functions, the complementary function concept underlines the necessity of determining these base solutions from which to build a general solution.

Wronskian Method

The Wronskian method is a technique used to determine whether a set of solutions forms a fundamental set for the space of solutions, specifically whether two functions are linearly independent. For two functions, \( y_1(x) \) and \( y_2(x) \), 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 \( W(y_1,y_2) eq 0 \) for all \( x \), then \( y_1(x) \) and \( y_2(x) \) are linearly independent and form a fundamental solution set. In our problem, this method indicates whether \( y_1(x) \) and \( y_2(x) \) can be used to describe the entire solution space for the given differential equation.

While the practical application in this problem becomes complex, understanding the role of the Wronskian gives insight into the structure of differential equation solutions. It's especially important when initial conditions are employed to find specific constants in solutions.