Question

Find the general solution of the given differential equation. $$ 12 y^{\mathrm{iv}}+31 y^{\prime \prime \prime}+75 y^{\prime \prime}+37 y^{\prime}+5 y=0 $$

Short answer

Answer: The general solution for a 4th-degree homogeneous linear differential equation with constant coefficients is given by: $$ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} + C_3 e^{r_3 x} + C_4 e^{r_4 x} $$ where \(r_1\), \(r_2\), \(r_3\), and \(r_4\) are the roots of the characteristic equation, and \(C_1\), \(C_2\), \(C_3\), and \(C_4\) are constant coefficients dependent on initial or boundary conditions.

Step-by-step solution

Write the characteristic equation

First, form the characteristic equation by replacing the derivatives with powers of \(r\). This will be a polynomial equation in \(r\): $$ 12r^4 + 31r^3 + 75r^2 + 37r + 5 = 0 $$

Factor the characteristic equation

The next step would typically be to factor the characteristic equation, but unfortunately, this equation does not factor easily. We will need to use numerical methods to find its roots.

Find the roots of the characteristic equation

Using numerical methods (such as the Newton-Raphson method or a root-finding software), find the four roots of the characteristic equation: $$r_1, r_2, r_3, r_4.$$

Write the general solution

Given these roots, we can write the general solution of the differential equation as follows: $$ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} + C_3 e^{r_3 x} + C_4 e^{r_4 x} $$ Where \(C_1\), \(C_2\), \(C_3\), and \(C_4\) are constant coefficients that would depend on initial or boundary conditions.

Key concepts

Characteristic Equation

In the realm of differential equations, a characteristic equation is a crucial tool for solving higher order differential equations. It is formed by replacing derivatives in the differential equation with powers of a new variable, usually denoted as \(r\). This transformation is key because it turns our complex differential equation into a polynomial equation. For example, consider the differential equation given in the problem where derivatives are replaced and results in a polynomial: \(12r^4 + 31r^3 + 75r^2 + 37r + 5 = 0\). The characteristic equation allows us to understand the behavior of solutions to the original differential equations by analyzing this polynomial. By finding the roots of this polynomial, we can determine the base solutions needed for composing the general solution of the differential equation. Understanding and forming a characteristic equation is the first essential step in tackling higher order differential equations.Use the characteristic equation to:

• Simplify solving complex differential equations by converting them to algebraic form.
• Identify possible types of solutions based on the roots.
• Proceed with numerical or analytical methods for root finding.

Numerical Methods

Sometimes, we encounter characteristic equations that do not factor neatly. When this happens, numerical methods become particularly vital. Numerical methods refer to a variety of strategies used to approximate solutions to mathematical equations that are otherwise difficult to solve analytically.For our characteristic equation \(12r^4 + 31r^3 + 75r^2 + 37r + 5 = 0\), exact algebraic solutions for the roots might be impossible due to complexity. Here’s where numerical methods come into play. Common Numerical Methods:

• Newton-Raphson Method: A widely-used numerical method that approximates roots by iteration. It is powerful but may require a good initial guess to converge effectively.
• Root-finding Software: Software tools and programming languages exist specifically for tackling these issues, providing ease of computation and accuracy.

These methods include steps to iterate toward an answer by progressively approximating the roots until a satisfactory solution is found, usually within a defined tolerance level. Implementing numerical methods allows us to continue solving even the most stubborn polynomial equations that resist straightforward factorization or analytical solutions.

General Solution

Once the roots of a characteristic equation are found, the next step is to construct the general solution of the differential equation. The general solution is a combination of solutions associated with each root of the characteristic equation.For our differential equation, the roots \(r_1, r_2, r_3,\) and \(r_4\) contribute to the general solution expressed in the form:\[y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} + C_3 e^{r_3 x} + C_4 e^{r_4 x}\] Here’s how it works:

• Each root \(r_i\) (where \(i\) is an index for roots) corresponds to a specific part of the solution, \(e^{r_i x}\).
• The coefficients \(C_1, C_2, C_3,\) and \(C_4\) are constants determined by initial conditions or boundary values.

This form of solution is leveraged because the exponential function \(e^{r x}\) captures not only the growth or decay behaviors but also oscillatory solutions if complex roots arise. The foundational concept here is the linear combination of solutions corresponds to the degree of the differential equation.

Roots of Polynomial Equations

Understanding the roots of polynomial equations is foundational in solving differential equations via the characteristic equation. These roots represent the solutions to the characteristic equation, but more importantly, they define the behavior of the solutions to the differential equation.Each root identifies an independent solution to the differential equation. The nature of these roots—whether real, complex, distinct, or repeated—determines the form of the general solution:

• Real Roots: Each gives rise to a simple exponential solution \(e^{rx}\).
• Complex Roots: Lead to oscillatory solutions and appear in conjugate pairs, contributing solutions akin to sine and cosine functions.
• Repeated Roots: These require multiplication by \(x\) to account for each repetition, modifying the exponential solutions appropriately.

These details are crucial, as they influence how the variable elements combine to meet initial conditions specified in real-world problems. Ultimately, understanding these roots allows for the precise composition of solutions that accurately represent the system described by the original differential equation.