Question

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution? $$ \sin (x) y^{\prime}+\cos (x) y=2 x $$

Short answer

Partial results depend on initial conditions: for some, the behavior changes at \( x = n\pi \).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is \( \sin(x) y' + \cos(x) y = 2x \). This is a first-order linear differential equation because it can be expressed in the standard form \( y' + P(x)y = Q(x) \), where \( P(x) = \frac{\cos(x)}{\sin(x)} \) and \( Q(x) = \frac{2x}{\sin(x)} \).

Calculate the Integrating Factor

To solve the differential equation, we calculate the integrating factor \( \mu(x) \) using the formula \( \mu(x) = e^{\int P(x) \, dx} \). Here, \( P(x) = \frac{\cos(x)}{\sin(x)} = \cot(x) \). Then, the integrating factor becomes \( \mu(x) = e^{\int \cot(x) \, dx} = e^{\ln|\sin(x)|} = |\sin(x)| \).

Rewrite the Differential Equation Using the Integrating Factor

We multiply the original differential equation by the integrating factor to simplify it: \( |\sin(x)| y' + |\sin(x)| \frac{\cos(x)}{\sin(x)} y = 2x |\sin(x)| \). This simplifies to \( |\sin(x)| y' + \cos(x) y = 2x |\sin(x)| \).

Integrate Both Sides

Recognizing that the left side is the derivative of \( (|\sin(x)| y) \), the equation is \( \frac{d}{dx}(|\sin(x)| y) = 2x |\sin(x)| \). We integrate both sides with respect to \( x \): \( |\sin(x)| y = \int 2x |\sin(x)| \, dx \).

Solve the Integral

Finding the integral, \( \int 2x |\sin(x)| \, dx \), the integral can be simplified or solved numerically. However, for practical purposes in plotting, assume this results in a function \( F(x) \). Thus, the integrated equation is \( |\sin(x)| y = F(x) + C \), where \( C \) is the constant of integration.

Analyze the Family of Solutions

The general solution is \( y = \frac{F(x) + C}{|\sin(x)|} \). Use a calculator or software to plot this family for various values of \( C \). Observe changes with different initial conditions, noticing if solutions exhibit different behaviors based on \( C \). In particular, look for values of \( x \) where \( \sin(x) = 0 \) since this will affect the solution domains.

Key concepts

First-Order Linear Differential Equations

First-order linear differential equations are equations that involve the first derivative of an unknown function. They have a standard form that can be expressed as:

• \( y' + P(x)y = Q(x) \)

Here, \( y' \) represents the derivative of the function \( y \) with respect to \( x \), and \( P(x) \) and \( Q(x) \) are continuous functions of \( x \).
This equation type is "linear" because the dependent variable and its derivative appear to the first power and are not multiplied together or composed.A key to identifying a first-order linear differential equation is to recognize that it can be restructured to this specific form. Once in standard form, we can deploy tools like an integrating factor to find its solutions.

Integrating Factor

An integrating factor is a vital tool for solving first-order linear differential equations. It transforms such an equation into a form where we can integrate both sides easily.
The integrating factor \( \mu(x) \) is typically defined as:

• \( \mu(x) = e^{\int P(x) \, dx} \)

Here, \( P(x) \) is the function found in the standard form of the differential equation \( y' + P(x)y = Q(x) \).In our given equation, \( P(x) \) happens to be \( \frac{\cos(x)}{\sin(x)} \) or \( \cot(x) \). Calculating the integrating factor involves evaluating the integral of \( \cot(x) \), which results in \( \mu(x) = |\sin(x)| \).
Applying this factor simplifies the original equation, thus allowing for straightforward integration.

Initial Conditions

Initial conditions are specific values provided for the function and its derivatives at a particular point. They help to determine a unique solution from the family of solutions of a differential equation.
For example, if you're given an initial condition like \( y(x_0) = y_0 \), you can use it to find the exact value of the constant \( C \) after integrating.Initial conditions are crucial when multiple solutions are possible because they 'pin down' one specific solution that fits the problem's requirements.
Without them, our solution would remain general, represented with a constant \( C \) indicating many possible solutions.

Solution Behavior

Understanding the solution behavior of a differential equation involves examining how it reacts under varying initial conditions and different parameter values.
For the given equation, plotting the family of solutions shows how changes in the constant of integration \( C \) affect the solution.Noticeably, the values of \( x \) where \( \sin(x) = 0 \) become important, as these points could lead to undefined behavior in the solutions, dividing the domain or affecting continuity.
This makes recognizing critical points where the function's behavior might alter essential. By adjusting \( C \) through various initial conditions, you can observe distinct solution paths and identify significant behavior changes.It is this adaptability and insight into the solution's behavior that enables one to predict how the system represented by the differential equation would respond to different scenarios.