Problem 57 - Advanced Engineering Mathematics

Question

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} x+c_{3} e^{7 x} $$

Step-by-Step Solution

Answer

The differential equation is \( \frac{d^3 y}{dx^3} - 7 \frac{d^2 y}{dx^2} = 0 \).

Step 1: Identify the form of the general solution

The given general solution is a combination of a polynomial and an exponential term: \( y = c_1 + c_2 x + c_3 e^{7x} \). This form suggests that we have a differential equation whose solutions include constant, linear, and exponential components.

Step 2: Analyze the structure of the solution

The terms \( c_1 \) and \( c_2 x \) imply that \( r=0 \) is a root with multiplicity 2 (since they represent solutions to a linear homogeneous equation with constant coefficients). The term \( c_3 e^{7x} \) suggests \( r=7 \) is another distinct root of the characteristic equation.

Step 3: Construct the characteristic equation

Based on the identified roots, the characteristic equation is constructed as: \((r-0)^2(r-7) = r^2(r-7)\). Therefore, the characteristic equation is \( r^3 - 7r^2 = 0 \).

Step 4: Establish the differential equation

The characteristic equation \( r^3 - 7r^2 = 0 \) corresponds to the homogeneous linear differential equation: \[ D^3y - 7D^2y = 0 \] where \( D \) is the differential operator \( \frac{d}{dx} \). Thus, the differential equation is \( \frac{d^3 y}{dx^3} - 7 \frac{d^2 y}{dx^2} = 0 \).

Key Concepts

Constant Coefficients

In the context of homogeneous linear differential equations, constant coefficients refer to the fixed values that multiply the derivatives of a function in a differential equation. These coefficients remain the same, or constant, throughout the entire equation. For example, in the differential equation \( \frac{d^3 y}{dx^3} - 7 \frac{d^2 y}{dx^2} = 0 \), the coefficients are 1 and -7, as they are the constant factors that appear before the third and second derivatives, respectively.

• "Constant" means these values do not change as the variable \( x \) changes.
• "Coefficients" are the numbers multiplying the derivatives of \( y \).

These constant coefficients simplify the process of finding and analyzing the solutions to the differential equation. They allow us to apply specific mathematical techniques, like creating a characteristic equation, to solve the differential equation efficiently.

Characteristic Equation

The characteristic equation is a crucial concept when working with homogeneous linear differential equations with constant coefficients. It is essentially a polynomial equation, and its roots help determine the form of the solution to the differential equation.To derive the characteristic equation from a differential equation like \( \frac{d^3 y}{dx^3} - 7 \frac{d^2 y}{dx^2} = 0 \), we substitute each differential term \( D^n y \) with \( r^n \). Here, \( D \) stands for the differential operator \( \frac{d}{dx} \), and \( r \) is a placeholder for roots of the equation.

• The given differential equation translates to the characteristic equation \( r^3 - 7r^2 = 0 \).
• The roots \( r = 0 \) with multiplicity 2 and \( r = 7 \) satisfy this polynomial equation.

These roots provide us with the necessary information to construct the general solution of the original differential equation.

General Solution

The general solution of a homogeneous linear differential equation is a combination of functions, each corresponding to the roots of the characteristic equation. For our differential equation \( \frac{d^3 y}{dx^3} - 7 \frac{d^2 y}{dx^2} = 0 \), we determined the roots to be \( r = 0 \) with multiplicity 2 and \( r = 7 \).Each root will generate a specific type of term in the general solution:

• For \( r = 0 \) with multiplicity 2, the terms are a constant \( c_1 \) and a linear monomial \( c_2 x \).
• For \( r = 7 \), the term is the exponential function \( c_3 e^{7x} \).

These components are combined to form the overall general solution: \( y = c_1 + c_2 x + c_3 e^{7x} \).This solution provides a complete description of the behavior of the system described by the differential equation, encompassing all possible solutions based on initial conditions.