Question

Use the principle of superposition to find a particular solution. Where indicated, solve the initial value problem. $$ y^{\prime \prime}-4 y^{\prime}+4 y=e^{2 x}(1+x)+e^{2 x}(\cos x-\sin x)+3 e^{3 x}+1+x $$

Short answer

Answer: The particular solution of the given second-order nonhomogeneous linear differential equation is \(y_p(x) = y_{p1}(x) + y_{p2}(x) + y_{p3}(x) + y_{p4}(x)\), where \(y_{p1}(x)\), \(y_{p2}(x)\), \(y_{p3}(x)\), and \(y_{p4}(x)\) are the particular solutions corresponding to the forcing functions \(e^{2x}(1+x)\), \(e^{2x}(\cos x - \sin x)\), \(3e^{3x}\), and \(1+x\), respectively.

Step-by-step solution

Identify Forcing Functions

First, we need to notice that there are four forcing functions in the given equation. They are: 1. \(e^{2x}(1+x)\) 2. \(e^{2x}(\cos x - \sin x)\) 3. \(3e^{3x}\) 4. \(1+x\) Now we will find the corresponding particular solutions for each forcing function.

Find Particular Solution for the First Forcing Function

For the first forcing function \(e^{2x}(1+x)\), we can assume a particular solution of the form: $$ y_p(x) = (A x + B)e^{2x} $$ where A and B are constants to be determined. Substitute the assumed particular solution into the equation and solve for the constants A and B. The resulting particular solution can be denoted as \(y_{p1}(x)\).

Find Particular Solution for the Second Forcing Function

For the second forcing function \(e^{2x}(\cos x - \sin x)\), we can assume a particular solution of the form: $$ y_p(x) = e^{2x}(C\cos x + D\sin x) $$ where C and D are constants to be determined. Substitute the assumed particular solution into the equation and solve for the constants C and D. The resulting particular solution can be denoted as \(y_{p2}(x)\).

Find Particular Solution for the Third Forcing Function

For the third forcing function \(3e^{3x}\), we can assume a particular solution of the form: $$ y_p(x) = E e^{3x} $$ where E is a constant to be determined. Substitute the assumed particular solution into the equation and solve for the constant E. The resulting particular solution can be denoted as \(y_{p3}(x)\).

Find Particular Solution for the Fourth Forcing Function

For the fourth forcing function \(1+x\), we can assume a particular solution of the form: $$ y_p(x) = Fx + G $$ where F and G are constants to be determined. Substitute the assumed particular solution into the equation and solve for the constants F and G. The resulting particular solution can be denoted as \(y_{p4}(x)\).

Use the Principle of Superposition

Now that we have found the particular solutions \(y_{p1}(x)\), \(y_{p2}(x)\), \(y_{p3}(x)\), and \(y_{p4}(x)\) for each forcing function, we can use the principle of superposition to find the overall particular solution. The particular solution \(y_p(x)\) is given by the sum of the individual particular solutions: $$ y_p(x) = y_{p1}(x) + y_{p2}(x) + y_{p3}(x) + y_{p4}(x) $$ Now, combine the particular solutions found in steps 2 to 5 to obtain the overall particular solution for the given differential equation.

Key concepts

Differential Equations

A differential equation is an equation that relates a function with its derivatives. These equations express how a function and its rate of change are interlinked.
A common form of differential equations is of the type: \( y'' + p(x)y' + q(x)y = f(x) \), where the function \( f(x) \) is known as the forcing function. Differential equations can model various physical phenomena, such as motion, heat, and waves.
When solving differential equations, determining the appropriate solution form is crucial. Solutions can be general, containing arbitrary constants, or particular, addressing specific conditions or inputs.
The equation presented in the exercise involves a second-order differential equation, where we're tasked to find particular solutions using the principle of superposition.

Forcing Functions

Forcing functions in differential equations are terms that drive the system's behavior. They act like sources of external influence. In our equation, they are on the right-hand side:

• \( e^{2x}(1+x) \)
• \( e^{2x}(\cos x - \sin x) \)
• \( 3e^{3x} \)
• \( 1+x \)

Each forcing function contributes differently to the solution. By analyzing them individually, we create tailored particular solutions that account for each separate influence.
This method aids in understanding how each function impacts the overall response of the system.

Particular Solution

A particular solution to a differential equation is a specific solution that fits the non-homogeneous equation given its forcing functions.
To find a particular solution, we choose a plausible function form based on the structure of the forcing function. For example, if the forcing function involves terms like \( e^{2x} \), we guess a solution involving similar terms.
Once guesses are made, these functions are substituted back into the original differential equation. By simplifying and matching coefficients, we solve for the unknown constants, yielding concrete particular solutions.
The ability to synthesize particular solutions effectively helps in constructing a complete solution that satisfies dynamic conditions.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation accompanied by specified values at a starting point.
It answers the question: What trajectory does the solution follow given a starting condition?
IVPs necessitate additional information, such as \( y(0) \) or \( y'(0) \), to determine a unique solution among infinite possibilities of general solutions.
Addressing an IVP involves not only solving for a general solution but modifying it to exactly meet the initial conditions, often combining with particular solutions.
This exercise didn't provide initial conditions, but typically, once an overall solution is found, IVPs ensure the solution describes the system's response at the onset of observation.