Question

Solve the generic equation \(y^{\prime}=a x+y .\) How does varying \(a\) change the behavior?

Short answer

Varying \(a\) changes the slope of the linear term in the solution \(y = Ce^x + \frac{a}{2}x\). Higher \(a\) leads to a steeper linear growth.

Step-by-step solution

Understand the Equation

The given equation is a first-order linear differential equation of the form \(y' = ax + y\). It means that the rate of change of \(y\) is determined by \(ax + y\). The parameter \(a\) is a constant that will affect the behavior of the solutions.

Solve Homogeneous Part

First, we solve the homogeneous part of the equation: \(y' = y\). Its solution is \(y_h = Ce^x\), where \(C\) is the constant of integration.

Find Particular Solution

For the particular solution \(y_p\), assume a form \(y_p = bx\). Substitute into the equation, \(b = a - b\), solving gives \(b = \frac{a}{2}\). So, \(y_p = \frac{a}{2}x\).

General Solution

Combine the homogeneous and particular solutions to get the general solution: \(y = Ce^x + \frac{a}{2}x\). This form shows how the solution behaves based on different values of \(a\).

Analyze the Role of Parameter \(a\)

The parameter \(a\) affects the slope of the particular solution \(\frac{a}{2}x\). A larger value of \(a\) increases the slope of this term, causing the overall solution to grow more steeply post the exponential effect.

Key concepts

First-Order Linear Differential Equations

In mathematics, differential equations are equations that involve a function and its derivatives. A first-order linear differential equation is one of the simplest forms and is expressed in the format \[ y' = ay + b(x) \] where \( y' \) represents the derivative of the function \( y(x) \), \( a \) is a constant, and \( b(x) \) is a function of \( x \). In the specific form we've been given, \[ y' = ax + y \]

• The equation is "linear" because the function \( y \) and its derivative \( y' \) appear to the first power, and there's no multiplication of these terms.
• The "first-order" descriptor refers to the highest derivative in the equation being the first derivative.

First-order linear differential equations are foundational in many areas of mathematics because they model a wide range of physical and natural phenomena, from radioactive decay to population dynamics.

Homogeneous Equation

A homogeneous differential equation is one where all terms involving the dependent variable and its derivatives can be set equal to zero. In simpler terms, it looks like this:\[ y' = ay \]The homogeneous counterpart of the given differential equation \( y' = ax + y \) is \( y' = y \). Solving a homogeneous equation typically involves finding a function that, when differentiated, reproduces the original equation's structure.

• In our case, the solution is the exponential function \( y_h = Ce^x \), where \( C \) is a constant determined by initial or boundary conditions.
• This solution represents how the system behaves when external forces are removed (i.e., when it is left to its intrinsic properties).

Since exponential functions grow rapidly, the homogeneous solution often dominates whenever the specific external influences are negligible compared to it.

Particular Solution

A particular solution to a differential equation is a solution that satisfies the non-homogeneous part of the equation. It reflects the effect of external input or non-zero terms in the equation. For the equation \( y' = ax + y \), we hypothesize a particular solution:\[ y_p = bx \]By substituting \( y_p = bx \) into the equation and solving for \( b \), we find that \( b = \frac{a}{2} \). This gives us the particular solution:\[ y_p = \frac{a}{2}x \]

• This form accommodates the specific term \( ax \) in the differential equation.
• The particular solution provides insight into how different values of \( a \) influence the linear part of the solution.

Understanding particular solutions is essential because they account for the specific conditions or inputs related to a problem.

General Solution

The general solution of a differential equation combines both the homogeneous solution and the particular solution. It represents the complete set of solutions considering both the natural behavior of the system and the effects of any external influences.For the equation \( y' = ax + y \), we have:

• The homogeneous solution: \( y_h = Ce^x \)
• The particular solution: \( y_p = \frac{a}{2}x \)

Combining these, the general solution is:\[ y = Ce^x + \frac{a}{2}x \]This general solution helps describe the slopes and curvatures exhibited by the solutions graphically.

• The term \( Ce^x \) dominates for larger values of \( x \) because exponential growth is very rapid.
• The linear term \( \frac{a}{2}x \) adjusts the slope based on the magnitude of \( a \), showing how quickly the solution changes in response to the linear component \( ax \).

General solutions are fundamental for predicting future behavior of systems modeled by differential equations, as they provide a versatile tool for understanding complex phenomena.