Question

Find the particular integral of each of the following equations by the method of undetermined coefficients: (a) \(y^{n}(t)+2 y^{\prime}(t)+y=t\) (b) \(y^{\prime \prime}(t)+4 y^{\prime}(t)+y=2 t^{2}\) (c) \(y^{\prime \prime}(t)+y^{\prime}(t)+2 y=e^{t}\) (d) \(y^{\prime \prime}(t)+y^{\prime}(t)+3 y=\sin t\)

Short answer

For each equation, choose a trial function based on the non-homogeneous term, substitute and solve for coefficients.

Step-by-step solution

Identify the Differential Equation Type

The method of undetermined coefficients is applicable to linear constant-coefficient differential equations where the non-homogeneous part is a sum of polynomial, exponential, sine, or cosine functions.

Choose a Trial Function for Equation (a)

For the equation (a) \(y^{n}(t)+2 y^{\prime}(t)+y=t\), the non-homogeneous term is \(t\). Therefore, we choose a trial particular integral of the form \(y_p(t) = At + B\).

Substitute the Trial Function into Equation (a)

Differentiate the trial function: \(y_p'(t) = A\). Substitute \(y_p(t) = At + B\) into the left side of the equation and equate to \(t\): \((At + B)^{n} + 2A = t\). Determine the coefficients \(A\) and \(B\) by equating the coefficients of like terms.

Repeat for Equation (b)

For \(y^{\prime \prime}(t)+4 y^{\prime}(t)+y=2 t^{2}\), the non-homogeneous term is \(2t^2\). Choose a trial function \(y_p = At^2 + Bt + C\). Substitute and equate coefficients to solve for \(A\), \(B\), and \(C\).

Repeat for Equation (c)

For \(y^{\prime \prime}(t)+y^{\prime}(t)+2 y=e^{t}\), the non-homogeneous term is \(e^{t}\). A suitable trial function is \(y_p = Ae^t\). Differentiate, substitute, and solve for \(A\).

Repeat for Equation (d)

For \(y^{\prime \prime}(t)+y^{\prime}(t)+3 y=\sin t\), the non-homogeneous term is \(\sin t\). Choose \(y_p = A\sin t + B\cos t\). Differentiate, substitute, and solve for \(A\) and \(B\).

Key concepts

Linear Differential Equations

Linear differential equations form the foundation for understanding more complex systems. They are characterized by the dependence of the derivative terms on the function, typically with constant coefficients. This means no powers or products of the function and its derivatives are involved. These equations can appear in various forms:

• First order: Involves only the first derivative, expressed as \( y' + p(t)y = g(t) \).
• Higher order: Involves second or higher derivatives, like \( y'' + ay' + by = f(t) \).

When applying the method of undetermined coefficients, the linearity ensures the procedure is systematic. The non-homogeneous function on the right side helps to select an appropriate trial function for solving the equation. However, remember that this method is optimal for constant coefficients and when the non-homogeneous part is a simple polynomial, exponential, or trigonometric function.

Particular Integral

A particular integral (PI) is a specific solution of a differential equation that directly accounts for non-homogeneous parts. Unlike the general solution, which can represent an infinite set of solutions with arbitrary constants, the particular integral captures the unique part contributed by the non-homogeneous term.

Finding the particular integral involves selecting an appropriate trial function resembling the form of the equation's non-homogeneous component. Let's see how it works:

• If the non-homogeneous term is a polynomial \( t^n \), the trial function might be \( At^n + Bt^{n-1} + \ldots \).
• For exponential terms like \( e^{mt} \), try \( Ae^{mt} \). If \( e^{mt} \) is part of the homogeneous solution, multiply by \( t \) to avoid duplication.
• For sine or cosine terms such as \( \sin(kt) \) or \( \cos(kt) \), use trial \( A\cos(kt) + B\sin(kt) \).

Once chosen, substitute this trial into the equation and solve for unknown coefficients to obtain the particular integral.

Homogeneous Equations

Homogeneous differential equations focus solely on the left-hand side of the equation without a non-homogeneous term. In mathematical terms, the equation takes the form \( L(y) = 0 \), where \( L(y) \) involves only the function \( y \) and its derivatives.

Such equations can be solved by assuming a solution structure like \( y = e^{rt} \), possibly leading to a characteristic equation. Solving this algebraic root equation, often a polynomial, determines the form of the homogeneous solution:

• Real and distinct roots \( r_1, r_2 \) yield solutions \( c_1e^{r_1t} + c_2e^{r_2t} \).
• Repeated roots \( r \) lead to solutions \( c_1e^{rt} + c_2te^{rt} \).
• Complex conjugate roots \( \alpha \pm i\beta \) result in \( e^{alpha t}(c_1\cos(\beta t) + c_2\sin(\beta t)) \).

The general solution combines the homogeneous solution with any particular integral to account for the non-homogeneous part, providing a comprehensive solution set for the equation.

Non-Homogeneous Equations

Non-homogeneous equations are those that include a function outside the homogeneous structure, expressed with \( L(y) = f(t) \), where \( f(t) \) is not zero. This presence of \( f(t) \) compels us to find not only a homogeneous solution but also a particular integral to account for \( f(t) \).

These equations often arise in practical scenarios, such as electrical circuits or mechanical vibrations, where an external force or source is present. When solving such equations:

• First, solve the homogeneous part \( L(y) = 0 \), as discussed in homogeneous equations.
• Next, derive the particular integral using a pragmatic trial function based on \( f(t) \), as outlined in the particular integral section.

The complete solution, representing both natural and external influences, is the sum of the homogeneous solution and the particular integral. It's this comprehensive approach that addresses both the natural dynamics and any external or source influences on the system.