Question

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

Short answer

Varying \(a\) affects the exponential growth rate in \(y = Ce^{\frac{a}{2}x^2} + f(x)\); positive \(a\) increases the growth exponentially while negative \(a\) diminishes it.

Step-by-step solution

Recognize the Given Differential Equation

The differential equation provided is of the form \(y^{\prime}=x+ax y\), where \(y'\) denotes the derivative of \(y\) with respect to \(x\). This is a first-order linear non-homogeneous differential equation.

Analyze the Effect of Varying Parameter 'a'

The parameter \(a\) affects the term \(axy\), which is a coupling between \(x\) and \(y\). Depending on the sign and magnitude of \(a\), this term can either amplify or diminish the relationship between \(x\) and \(y\), which will influence the slope \(y'\).

Determine Superposition of Homogeneous and Particular Solutions

To solve this, we find the general solution by adding the complementary solution (for the homogeneous equation) and a particular solution of the full equation. The homogeneous form \(y^{\prime} = ax y\) gives \(y_h\).

Solve the Homogeneous Equation

Separating variables for \(y^{\prime} = axy\), we have \( \frac{dy}{y} = ax \ dx \). Integrating both sides, \( \ln |y| = \frac{a}{2}x^2 + C \), leading to \( y_h = Ce^{\frac{a}{2}x^2} \), where \(C\) is the integration constant.

Solve the Particular Solution

Finding a particular solution to \(y^{\prime} = x + ax y\), assume \(y_p = mx + b\). Substituting into the equation and solving for \(m\) and \(b\) gives explicit expressions depending on initial or particular conditions.

Construct the General Solution

The general solution is of the form \( y = y_h + y_p = Ce^{\frac{a}{2}x^2} + f(x) \), where \(f(x)\) is the particular solution found in the previous step. This shows the complete behavior of \(y\) with respect to varying \(a\).

Analyze the Behavior as 'a' Varies

The exponential term \(e^{\frac{a}{2}x^2}\) indicates that for positive \(a\), \(y\) grows exponentially with increasing \(x\), while for negative \(a\), the growth diminishes. For \(a=0\), behavior depends solely on the particular solution.

Key concepts

First-Order Linear

A first-order linear differential equation involves terms where the highest order of the derivative is one. These equations typically take the form \( y' + P(x)y = Q(x) \), where \( y' \) is the derivative of \( y \) with respect to \( x \), \( P(x) \) and \( Q(x) \) are functions of \( x \), or simply constants. In our exercise, the equation is \( y' = x + axy \).
Here, \( y' \) represents the rate at which \( y \) changes with \( x \), making it a quintessential example of a first-order linear form. Understanding that these equations describe how one quantity changes with respect to another is key. They are "linear" because the function \( y \) and its derivative appear linearly (not raised to any power other than one).
Such equations are the foundation for more complex differential equations and often represent natural processes, like population growth or decay, where the rate of change depends linearly on the current state.

Homogeneous Solutions

The homogeneous solution of a differential equation is derived by setting \( Q(x) = 0 \) in the linear form \( y' + P(x)y = 0 \). This leads to solutions that depend solely on the function \( P(x) \).
In the equation \( y' = axy \), setting the non-homogeneous part to zero results in \( y' = axy \) itself because the only non-homogeneous part is the constant term \( Q(x) = x \).
By separating variables and integrating, we get \( \ln |y| = \frac{a}{2}x^2 + C \), leading to a homogeneous solution of \( y_h = Ce^{\frac{a}{2}x^2} \).
This part of the solution reflects the "internal" structure of the differential equation without external forcing terms, showing what the system behaves like from its intrinsic parameters alone.

Particular Solutions

A particular solution of a non-homogeneous differential equation directly satisfies the equation when specific functions (or constants from \( Q(x) \)) are involved. Unlike the homogeneous solution, it does not contain an arbitrary constant.
For the equation \( y' = x + axy \), we assume a trial solution form, say \( y_p = mx + b \). This is a function that does indeed make the entire differential equation valid when swapped in and potentially adapted.

Solving for \( m \) and \( b \) gives specific values based on conditions, akin to drawing a curve through a particular set of points. This particular solution gives insight into how the system responds to external influences or non-zero terms in \( Q(x) \). It shows the influence of initial conditions or external forcing on the solution.

Parameter Variation

Parameter variation is a crucial technique in understanding how changing a parameter in a differential equation affects the solution. In the context of \( y' = x + axy \), the parameter \( a \) plays a significant role.
This parameter influences the term \( axy \), altering how \( x \) and \( y \) interact. Changing \( a \) shifts the balance of influence from the non-homogeneous part \( x \) to the homogeneous coupling \( axy \).
For positive \( a \), as \( x \) increases, the solution \( y = Ce^{\frac{a}{2}x^2} + y_p \) tends to grow more steeply since the exponential component rises. If \( a \) is negative, the opposite occurs, with the exponential part potentially dampening the overall growth.
Through parameter variation, one sees how disparate system behaviors manifest under different conditions, crucial for fine-tuning models to better replicate real-life phenomena.