Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Show that if \(y_{p, 1}(t)\) solves \(y^{\prime \prime}(t)+p y^{\prime}(t)+q y(t)=f_{1}(t)\) and \(y_{p, 2}(t)\) solves \(y^{\prime \prime}(t)+p y^{\prime}(t)+q y(t)=f_{2}(t),\) then for any numbers \(A\) and \(B\) $$ A y_{p, 1}(t)+B y_{p, 2} \quad \text { solves } \quad y^{\prime \prime}(t)+p y^{\prime}(t)+q y(t)=A f_{1}(t)+B f_{2}(t) $$

Step-by-Step Solution

Answer

A linear combination \( Ay_{p,1}(t) + By_{p,2}(t) \) solves the equation with \( Af_1(t) + Bf_2(t) \) as the forcing function.

Step 1: Identify the Homogeneous Solution

The given differential equations have the form of a second-order linear differential equation:\[ y''(t) + p y'(t) + q y(t) = f(t) \]Where the right side of the equation is a function, \( f(t) \). A particular solution \( y_{p}(t) \) solves this equation for specific \( f(t) \). Our task involves showing that a linear combination of two particular solutions, \( y_{p,1}(t) \) and \( y_{p,2}(t) \), for different right side functions, solves a new equation with a combined right side. This doesn't involve the homogeneous part directly but understanding that \( y(t) = y_h(t) + y_p(t) \), where \( y_h(t) \) is the homogeneous solution, makes linearity clear.

Step 2: Verify Particular Solutions

Ensure that \( y_{p,1}(t) \) satisfies:\[ y''_{p,1}(t) + p y'_{p,1}(t) + q y_{p,1}(t) = f_1(t) \]and \( y_{p,2}(t) \) satisfies:\[ y''_{p,2}(t) + p y'_{p,2}(t) + q y_{p,2}(t) = f_2(t) \]These are given and will form the basis for constructing the solution to the combined equation.

Step 3: Construct a New Particular Solution

Formulate a new particular solution as a linear combination:\[ y_{p}(t) = A y_{p,1}(t) + B y_{p,2}(t) \]Here, \( A \) and \( B \) are constants that scale the contributions of the solutions \( y_{p,1}(t) \) and \( y_{p,2}(t) \).

Step 4: Apply the Operator to the Linear Combination

Apply the differential operator \( D^2 + pD + q \) to \( y_p(t) \):\[ (A y_{p,1}''(t) + B y_{p,2}''(t)) + p(A y_{p,1}'(t) + B y_{p,2}'(t)) + q(A y_{p,1}(t) + B y_{p,2}(t)) \]This simplifies to:\[ A(y''_{p,1}(t) + p y'_{p,1}(t) + q y_{p,1}(t)) + B(y''_{p,2}(t) + p y'_{p,2}(t) + q y_{p,2}(t)) \]

Step 5: Substitute Particular Solutions into the Equation

Substitute the individual equations for \( y_{p,1}(t) \) and \( y_{p,2}(t) \) into the expression obtained in Step 4:\[ A f_1(t) + B f_2(t) \]since from Step 2, we've confirmed:\[ y''_{p,1}(t) + p y'_{p,1}(t) + q y_{p,1}(t) = f_1(t) \]and\[ y''_{p,2}(t) + p y'_{p,2}(t) + q y_{p,2}(t) = f_2(t) \]Therefore, our combined terms result in:\[ A f_1(t) + B f_2(t) \]

Step 6: Conclude the Solution

We have shown that the linear combination \( A y_{p,1}(t) + B y_{p,2}(t) \) satisfies:\[ y''(t) + p y'(t) + q y(t) = A f_1(t) + B f_2(t) \]Thus, the linearity of the differential operator implies that if individual solutions of a linear differential equation can be scaled and added together, then their linear combination is a solution to the equation with the scaled and added functions on the right side.

Key Concepts

superposition principle

The superposition principle is a fundamental concept in the study of linear differential equations. It states that if two functions are solutions to a linear differential equation, their linear combination is also a solution to the same equation. This principle greatly simplifies the process of solving differential equations, especially when dealing with complex systems.
To illustrate, consider functions \( y_{p,1}(t) \) and \( y_{p,2}(t) \) that are solutions to their respective equations \( y''(t) + py'(t) + qy(t) = f_1(t) \) and \( y''(t) + py'(t) + qy(t) = f_2(t) \). According to the superposition principle, a combination like \( Ay_{p,1}(t) + By_{p,2}(t) \), with \( A \) and \( B \) as constants, will solve the equation \( y''(t) + py'(t) + qy(t) = Af_1(t) + Bf_2(t) \).
This is useful because it allows you to break down a complex problem into simpler parts, solve each part separately, and combine the solutions. This approach is not limited to second-order differential equations but applies generally to any linear system.

particular solutions

Particular solutions are specific solutions that address the non-homogeneous part of a differential equation. When dealing with a linear differential equation, the general solution comprises two components: the homogeneous solution and the particular solution.
Consider a non-homogeneous differential equation of the form \( y''(t) + py'(t) + qy(t) = f(t) \). A particular solution \( y_p(t) \) satisfies this equation exclusively for the non-homogeneous part \( f(t) \).
For example, if \( y_{p,1}(t) \) solves \( y''(t) + py'(t) + qy(t) = f_1(t) \) and \( y_{p,2}(t) \) solves \( y''(t) + py'(t) + qy(t) = f_2(t) \), each is a particular solution for its respective function. These solutions focus solely on matching the right side of the equation and ignore any initial or boundary conditions unless specifically given.
Understanding particular solutions allows us to tackle custom parts of differential equations effectively. By focusing on the non-homogeneous part, particular solutions pave the way to constructing the full solution by adding them to the homogeneous solution.

linearity of differential operators

The linearity of differential operators is key to understanding how superposition and combining particular solutions work in the realm of differential equations. A differential operator is said to be linear if it satisfies two properties: scaling and additivity.
Firstly, scaling implies that for a constant \( C \), applying the operator to \( Cy(t) \) is the same as applying the constant times the operator to \( y(t) \): \[ L[Cy(t)] = C L[y(t)] \] This means that if you scale a function by a constant, the transformation by the operator scales by the same constant.
Secondly, additivity means that the operator applied to the sum of functions is the sum of the operator applied to each function: \[ L[y_1(t) + y_2(t)] = L[y_1(t)] + L[y_2(t)] \] This ensures that individual solutions to a differential equation can be combined into a single solution.
In reference to our exercise, the linearity allows us to confidently state that if \( y_{p,1}(t) \) and \( y_{p,2}(t) \) are solutions to differential equations with different functions on the right side, their linear combination is a valid solution to a new equation formed by the linear combination of those right-side functions. This guarantees that combining solutions is mathematically sound and reliable.