Problem 37 - Differential Equations and Linear Algebra

Question

State whether the annihilator method can be used to determine a particular solution to the given differential equation. If the technique cannot be used, state why not. If the technique can be used, then give an appropriate trial solution. $$y^{\prime \prime}-2 y^{\prime}+5 y=7 e^{x} \cos x+\sin x.$$

Step-by-Step Solution

Answer

The annihilator method can be used for the given differential equation. The trial solution for the particular solution is: \(y_p(x) = e^x(C\cos x + D\sin x) + E\cos x + F\sin x\), where \(C, D, E\), and \(F\) are constants to be determined. The complete solution will be of the form: \(y(x) = y_c(x) + y_p(x)\), where \(y_c(x) = e^x(A\cos(2x) + B\sin(2x))\).

Step 1: Identify the homogeneous part of the differential equation

The given differential equation is: \[\begin{cases} y^{\prime \prime}-2y^{\prime}+5y = 7e^x\cos x+\sin x \end{cases}\] The associated homogeneous part is: \(y^{\prime \prime}-2y^{\prime}+5y = 0\)

Step 2: Check if the annihilator method can be applied

We need to first check if we can apply the annihilator method to the given nonhomogeneous part of the differential equation. The annihilator method is applicable if we can find linear differential operators that will annihilate the given nonhomogeneous term. The term in question is \(7e^x\cos x+\sin x\).

Step 3: Identifying the appropriate annihilators

In this case, the nonhomogeneous term consists of two parts: \(7e^x\cos x\) and \(\sin x\). To annihilate \(7e^x\cos x\), we can use the operator \((D^2 - 2D + 5)(D^2 + 1)\), where \(D\) represents the derivative with respect to \(x\). This operator will annihilate \(e^x\cos x\) as it satisfies the homogeneous differential equation \((D^2 - 2D + 5)y = 0\), and \(D^2 + 1\) will annihilate any function of form \(a\sin x + b\cos x\) (where \(a\) and \(b\) are constants). Similarly, for the second part \(\sin x\), the appropriate annihilator is also \((D^2 + 1)\) as it annihilates any function of form \(a\sin x + b\cos x\). Since we can find suitable annihilators for both the nonhomogeneous terms, the annihilator method can be applied to the given differential equation.

Step 4: Construct the trial solution

To construct the trial solution using the annihilator method, we first find the complimentary solution \(y_c\) that corresponds to the given homogeneous equation \((y^{\prime \prime}-2y^{\prime}+5y=0)\). The characteristic equation is: \(r^2 - 2r + 5 = 0\) Using the quadratic formula, we find: \(r = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 5}}{2 \cdot 1} = 1 \pm 2i\) Since we have complex roots, the complementary solution will be of the form: \(y_c(x) = e^x(A\cos(2x) + B\sin(2x))\), where \(A\) and \(B\) are constants. For the particular solution, we need the following trial function taking into account the forms annihilated by \((D^2 - 2D + 5)(D^2 + 1)\): \(y_p(x) = e^x(C\cos x + D\sin x) + E\cos x + F\sin x\) , where \(C,D,E,F\) are constants to be determined. The complete solution will then be of the form: \(y(x) = y_c(x) + y_p(x)\)

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve derivatives of an unknown function. They are used to describe various applications such as motion, heat flow, electricity, and more by forming relationships between changing quantities.
These equations can be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs), depending on whether they involve derivatives concerning one or several variables.
Different techniques are employed to solve differential equations, including separation of variables, integrating factors, and the annihilator method. These methods help find solutions that satisfy the given differential equation.

Particular Solution

In the realm of differential equations, a particular solution refers to a specific solution of a nonhomogeneous differential equation. It uniquely satisfies the nonhomogeneous part of the equation, meaning the part that includes external forces or influence.
When solving a differential equation, both complementary and particular solutions are considered. The complementary solution addresses the homogeneous portion, while the particular solution deals with nonhomogeneous factors.
The general solution is the sum of both complementary and particular solutions, providing a comprehensive solution that satisfies the entire nonhomogeneous differential equation.

Trial Solution

A trial solution is an initial guess or form proposed to solve a nonhomogeneous differential equation, particularly when using methods like the annihilator method or variation of parameters.
The trial solution is based on the nature of the nonhomogeneous term. For instance, if the nonhomogeneous term involves trigonometric or exponential functions, the trial solution should incorporate these forms as well.
Coefficients within the trial solution are adjusted through substitution and comparison to derive constants. The trial solution is then integrated with the complementary solution to define a complete general solution for the original equation.

Nonhomogeneous Equations

A nonhomogeneous differential equation is distinguished from a homogeneous equation by the presence of an external, non-zero forcing function. This term is added to the differential equation and usually represents external forces or inputs in a system, like a target population, external pressure, or a driving force.
Nonhomogeneous equations are essential in modeling real-world scenarios where external forces or influences cannot be ignored. The solution involves both the homogeneous solution, which solves the corresponding homogeneous equation, and the particular solution that directly addresses the external inputs.
Techniques such as the annihilator method, undetermined coefficients, and variation of parameters are typically employed to find a particular solution, aiding to form the general solution of nonhomogeneous differential equations.