Question

(a) Determine a suitable form for \(Y(t)\) if the method of undetermined coefficients is to be used. (b) Use a computer algebra system to find a particular solution of the given equation. $$ y^{\prime \prime}+y=t(1+\sin t) $$

Short answer

Based on the solution provided, the particular solution for the given non-homogeneous differential equation \(y^{\prime \prime}+y=t(1+\sin t)\) using the method of undetermined coefficients is: $$ Y(t) = 2 \cos t + (t + 2) \sin t $$

Step-by-step solution

Identify the given differential equation

The given differential equation is: $$ y^{\prime \prime}+y=t(1+\sin t) $$

Determine the suitable form for Y(t)

We want to find a suitable form for Y(t) using the method of undetermined coefficients. The right-hand side of the equation is a product of a polynomial and a trigonometric function, i.e., \(t(1+\sin t)\) So, we will try to guess a solution that also consists of a polynomial and a trigonometric function: $$ Y(t) = (At + B) \cos t +(Ct + D) \sin t $$ where A, B, C, and D are the undetermined coefficients.

Calculate the first and second derivatives of Y(t)

Now, we need to find the first and second derivatives of Y(t) with respect to t. First derivative: $$ Y'(t) = - (At + B) \sin t + A \cos t + (Ct + D) \cos t + C \sin t $$ Second derivative: $$ Y''(t) = - (At + B) \cos t - 2A \sin t - (Ct + D) \sin t - 2C \cos t $$

Substitute Y(t), Y'(t), and Y''(t) into the given differential equation

Substitute Y(t), Y'(t), and Y''(t) into the given differential equation: $$ -(At + B) \cos t - 2A \sin t - (Ct + D) \sin t - 2C \cos t + (At + B) \cos t +(Ct + D) \sin t = t(1+\sin t) $$

Equate the coefficients

We now need to equate the coefficients of \(\cos t\), \(\sin t\), and the constants on both sides of the equation: (1) \(B - 2C = 0\) (2) \(A - 2A = 0\) (3) \(At + Ct = t\) (4) \(-Bt - Dt = 0\) Since \(A - 2A = 0\), \(A = 0\). From (1), we have \(B = 2C\). From (3), we can see that \(C = 1\). So, \(B = 2C = 2\). From (4), since \(-Bt - Dt = 0\), we have \(D = 2\). Now, we have A = 0, B = 2, C = 1, and D = 2.

Form the particular solution Y(t)

Substitute the found coefficients back into the assumed form of the particular solution Y(t): $$ Y(t) = 2 \cos t + (t + 2) \sin t $$ This is the particular solution of the given non-homogeneous differential equation using the method of undetermined coefficients. The use of a computer algebra system to solve the given equation would involve entering the differential equation, specifying the dependent and independent variables, and receiving the solution in the desired format. There are several computer algebra systems available, such as Mathematica, MATLAB, or Python libraries like SymPy.

Key concepts

Differential Equations

Differential equations are equations that involve unknown functions and their derivatives. They come up frequently in various disciplines, such as engineering, physics, and mathematics, because they can describe complex real-world systems.
A differential equation, like the given one \[y^{\prime \prime}+y=t(1+\sin t)\], has terms that include the derivatives of an unknown function, usually represented as \(y\). Here, it says the second derivative of \(y\), plus \(y\) itself, equals \(t(1+ \sin t)\).
**Types of Differential Equations**
There are different types of differential equations, but they're generally categorized as either ordinary or partial:

• **Ordinary Differential Equations (ODE):** Involves derivatives of a function with respect to a single variable.
• **Partial Differential Equations (PDE):** Involves derivatives with respect to multiple variables.

Within ordinary differential equations, further classifications exist, such as linear or nonlinear, depending on whether the function and its derivatives appear linearly. The given equation is a second-order linear ODE, as it involves the second derivative and the unknown \(y\) linearly.
Understanding differential equations and their solutions allows us to model and predict behaviors of systems like waves, heat distribution, or even population dynamics.

Particular Solution

A particular solution is a solution that satisfies a given non-homogeneous differential equation. This differs from the general solution, which includes arbitrary constants representing a family of solutions.
When using the method of undetermined coefficients, the particular solution is guessed based on the form of the forcing term on the right-hand side of the equation. In our example, the non-homogeneous part is \(t(1+\sin t)\), suggesting a suitable form for the particular solution as the combination of polynomial and trigonometric terms.
**Finding a Particular Solution**
Here's how you typically find it:

• **Form Selection:** Choose a trial solution form that mirrors the structure of the non-homogeneous term. Here, it was chosen as \((At + B) \cos t + (Ct + D) \sin t\).
• **Differentiate:** Calculate the derivatives needed for substitution into the differential equation.
• **Substitute and Solve:** Substitute the guessed form and its derivatives back into the original equation and solve for the undetermined coefficients by matching coefficients on both sides.

In this exercise, the modified equation is matched with the original to solve and find values for \(A = 0\), \(B = 2\), \(C = 1\), and \(D = 2\), leading to the particular solution: \[Y(t) = 2 \cos t + (t + 2) \sin t\].
This process gives a specific solution tailored to this equation, thus accomplishing the task of the particular solution.

Computer Algebra System

A computer algebra system (CAS) is software designed to perform symbolic mathematics. This includes algebraic operations like factoring, solving equations, differentiation, integration, and more. Examples of popular computer algebra systems include Mathematica, MATLAB, and Python's SymPy library.
**Advantages of Using CAS**
Employing a computer algebra system offers several advantages:

• **Efficiency:** Automates complex calculations that could be time-consuming and error-prone if done manually.
• **Accuracy:** Reduces the risk of human error in arithmetic and algebraic manipulation.
• **Versatility:** Supports a wide range of mathematical functions and utilities for various applications.

To use a CAS to find a particular solution to a differential equation, you would input the equation along with boundary conditions (if applicable). The system would process this input and return the solution in a desirable form. It simplifies the task of finding solutions analytically, which would otherwise take considerable manual effort.
In the context of this problem, utilizing a CAS helps verify the result obtained from the method of undetermined coefficients, ensuring consistency and correctness of the derived particular solution.