Problem 62 - Calculus

Question

Approach a steady state \(y_{x}\) as \(t \rightarrow \infty\). True or false, when \(y_{1}\) and \(y_{2}\) both satisfy \(y^{\prime}=c y+s\) (a) The sum \(y=y_{1}+y_{2}\) also satisfies this equation. (b) The average \(y=\frac{1}{2}\left(y_{1}+y_{2}\right)\) satisfies the same equation. (c) The derivative \(y=y_{1}^{\prime}\) satisfies the same equation.

Step-by-Step Solution

Answer

(a) False, (b) True, (c) False.

Step 1: Understand the problem

We need to determine if specific combinations of the solutions to the differential equation \( y' = cy + s \) also satisfy it. Given that \( y_1 \) and \( y_2 \) are solutions, we'll check if their sum, average, and derivative also satisfy the equation.

Step 2: Evaluate the sum y=y_{1}+y_{2}

Check if the sum \( y = y_1 + y_2 \) satisfies the differential equation \( y' = c y + s \):- The derivative of the sum is \( y' = y_1' + y_2' \).- Since \( y_1' = cy_1 + s \) and \( y_2' = cy_2 + s \), substituting these gives \( y' = (cy_1 + s) + (cy_2 + s) = c(y_1 + y_2) + 2s\).- Therefore, \( y' \) is not equal to \( c(y_1 + y_2) + s \) unless \( s = 0 \). Thus, this statement is False.

Step 3: Evaluate the average y=\frac{1}{2}(y_{1}+y_{2})

Check if the average \( y = \frac{1}{2}(y_1 + y_2) \) satisfies the differential equation:- The derivative of the average is \( y' = \frac{1}{2}(y_1' + y_2') \).- Substituting from the derivatives \( y_1' = cy_1 + s \) and \( y_2' = cy_2 + s \), we have \( y' = \frac{1}{2}(cy_1 + s + cy_2 + s) = c\frac{1}{2}(y_1 + y_2) + s \).- Therefore, this satisfies the equation, so this statement is True.

Step 4: Evaluate the derivative y=y_{1}^{\prime}

Determine if \( y = y_1' \) satisfies the differential equation:- For \( y = y_1' \), the differential equation would be \( y' = y_1'' \).- However, from the original differential equation \( y_1' = cy_1 + s \), we have \( y_1'' = c y_1' \).- Therefore, \( y' = y_1'' = cy_1' eq cy_1' + s \).- This means that \( y_1' \) does not satisfy the given equation unless \( s = 0 \), so this statement is False.

Key Concepts

steady state

In differential equations, the term "steady state" refers to a condition where the system's behavior becomes consistent over time, no longer changing as time continues to pass. In mathematical terms, when a system approaches a steady state, its derivative becomes zero as time approaches infinity. Analyzing whether a system reaches a steady state involves solving the differential equation and examining its behavior as time progresses. For example, if you have a solution to a differential equation, you can find out whether this solution approaches a steady state as time goes to infinity. This is crucial in fields like physics and engineering, where understanding the long-term stability of systems can be important. Steady states provide insight into the equilibrium behavior of systems, offering a glimpse into the eventual outcome of dynamic processes.

solutions to differential equations

Differential equations are mathematical equations that involve the derivatives of a function. The purpose of solving these equations is to find the function itself. Each differential equation can have many solutions, forming a family of functions that satisfy the equation. Finding solutions involves integrating the differential equation, sometimes using initial conditions to pinpoint a specific solution out of the family. In the exercise, both functions \(y_1\) and \(y_2\) are considered solutions to the differential equation \(y' = cy + s\). This means that each function satisfies the equation individually. Solving differential equations often requires techniques like separation of variables, integrating factors, or using specific boundary conditions to simplify and resolve the problem.

derivative of sum

The concept of the derivative of sum is an important rule in calculus. It states that the derivative of a sum of functions is equal to the sum of their derivatives. This rule simplifies complex derivative calculations by allowing us to handle each term of the sum separately. For example, if you have two functions \(f(x)\) and \(g(x)\), the derivative of their sum \(f(x) + g(x)\) is \(f'(x) + g'(x)\). In the context of the exercise, when calculating the derivative of the sum \(y = y_1 + y_2\), you find \(y' = y_1' + y_2'\). This simple but powerful rule is fundamental in analyzing how the sum of two solutions behaves in relation to a given differential equation.

linear combinations in differential equations

Linear combinations in differential equations involve the summation or weighted addition of solutions. It's a significant concept because linear systems often allow combinations of solutions to also be solutions to the equation. When we talk about linear differential equations, it means we can express the solution as a linear combination of simpler solutions. In the exercise, evaluating whether \(y = y_1 + y_2\) or \(y = \frac{1}{2}(y_1 + y_2)\) remain solutions highlights this idea. A linear combination like the average is a valid solution under certain conditions, specifically when scaling down maintains the same pattern of behavior dictated by the differential equation. Understanding how linear combinations function provides insights into how complex systems can be broken down into simpler, comprehensible components. This decomposition into linear combinations is widely used in fields such as linear algebra and systems analysis.