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? $$ x y^{\prime}+\frac{y}{2}=\sin (3 t) $$

Short answer

Solutions vary with the initial conditions due to changes in constant \(C\); zero initial value may lead to undefined solutions because of denominator terms.

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is \( x y^{\prime} + \frac{y}{2} = \sin(3t) \). This is a first-order linear differential equation in the form \( a(t)y' + b(t)y = c(t) \). In this case, \( a(t) = x \), \( b(t) = \frac{1}{2} \), and \( c(t) = \sin(3t) \).

Standard Form

The goal is to rewrite the equation in the standard linear form \( y^{\prime} + P(t) y = Q(t) \). Dividing the entire equation by \( x \), we get: \[ y^{\prime} + \frac{y}{2x} = \frac{\sin(3t)}{x} \]. Thus, \( P(t) = \frac{1}{2x} \) and \( Q(t) = \frac{\sin(3t)}{x} \).

Find the Integrating Factor

The integrating factor \( \mu(t) \) is given by \( e^{\int P(t) \ dt} \). Compute the integral: \[ \int \frac{1}{2x} \, dt = \frac{1}{2} \ln |x| = \ln |x|^{1/2} \]. Thus, the integrating factor is \( \mu(t) = e^{\ln |x|^{1/2}} = |x|^{1/2} \).

Apply the Integrating Factor

Multiply the entire differential equation by the integrating factor \( |x|^{1/2} \) to get: \[ |x|^{1/2} y^{\prime} + \frac{1}{2x} |x|^{1/2} y = \frac{\sin(3t)}{x} |x|^{1/2} \]. The left side becomes a derivative of a product: \[ \frac{d}{dt}(|x|^{1/2} y) = \frac{\sin(3t)}{x} |x|^{1/2} \].

Integrate Both Sides

Integrate both sides with respect to \(t\) to find \(y\). The left side is straightforward: \[ |x|^{1/2} y = \int \frac{\sin(3t)}{x} |x|^{1/2} \, dt + C \]. Assuming definite integral with respect to \(t\), the solution takes the form: \[ |x|^{1/2} y = -\frac{\cos(3t)}{3x} |x|^{1/2} + C \]. Simplify to solve for \(y\): \[ y = -\frac{\cos(3t)}{3x} + \frac{C}{|x|^{1/2}} \].

Analyze Initial Conditions

For different initial conditions \(y(0) = y_0\), you substitute to find a specific \(C\). Different \(C\) will shift the family of solutions up or down, thus changing the behavior of the solution graph. Check if specific conditions such as zero lead to undefined situations due to the presence of \(x\) in the denominator.

Key concepts

First-Order Linear Differential Equation

A first-order linear differential equation is a type of equation that involves the derivatives of a function and can be expressed in a standard form. Such equations typically involve a first derivative of a variable, which makes them one of the most fundamental types of differential equations you'll encounter. In the given problem, we are working with the equation \[ x y^{\prime} + \frac{y}{2} = \sin(3t) \]. This equation can be recognized as a first-order linear differential equation because:

• The highest derivative is the first derivative \(y'\).
• The equation is linear in its terms involving \(y\) and \(y'\).

To simplify this equation, we bring it to the standard linear form of a differential equation: \[ y^{\prime} + P(t) y = Q(t) \], where \(P(t)\) and \(Q(t)\) are functions of \(t\). By dividing the original equation by \(x\), the standard form becomes \[ y^{\prime} + \frac{y}{2x} = \frac{\sin(3t)}{x} \]. Understanding and transforming the equation into this form is crucial as it prepares the way for using other techniques, such as finding an integrating factor. This step sets the stage for solving the differential equation.

Integrating Factor

The integrating factor is an essential tool for solving first-order linear differential equations. It is a function that, when multiplied with the differential equation, allows the left-hand side to become the derivative of a product, making the equation easier to solve. For our exercise, the integrating factor \(\mu(t)\) is calculated from the function \(P(t)\), using the formula:\[\mu(t) = e^{\int P(t) \, dt}\]Where \(P(t) = \frac{1}{2x}\) for our transformed equation. Integrating gives:\[\int \frac{1}{2x} \, dt = \frac{1}{2} \ln |x| \]This simplifies the integrating factor to:\[\mu(t) = |x|^{1/2}\]By multiplying the entire differential equation by this integrating factor, the left-hand side results in a derivative of \(|x|^{1/2} y\). This simplification means the original differential equation can be rewritten effectively, dramatically easing the path to solving it. The integrating factor transforms the equation's complexity into a more manageable form, leading directly to the integration step.

Initial Conditions Analysis

In differential equations, initial conditions are essential as they allow you to determine a particular solution from a family of solutions. For the given differential equation, once solved, the general solution appears:\[y = -\frac{\cos(3t)}{3x} + \frac{C}{|x|^{1/2}}\]Here, the constant \(C\) signifies an arbitrary constant that can be determined by applying an initial condition. For example, if we have \(y(0) = y_0\), by substituting \(t = 0\) and \(y = y_0\) into the equation, we can solve for \(C\).

• Different values of \(C\) result in different curves or shifts in the family of solutions, directly affecting how the solution graph behaves.
• It's important to observe initial conditions carefully to avoid undefined behavior. For instance, values of \(x\) that cause divisions by zero must be carefully handled.

Understanding how initial conditions affect the differential equation solution is crucial, especially when predicting how solutions will behave under varied scenarios. Paying attention to these conditions helps in visualizing and interpreting the implications of different starting points for the function's behavior.