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? $$ (x+1) y^{\prime}=3 y+x^{2}+2 x+1 $$

Short answer

The solution family is \( y = -(x+1)^2 + C(x+1)^3 \). Initial conditions affect solution behavior via constant \( C \).

Step-by-step solution

Identify the Type of Differential Equation

First, observe whether the differential equation is separable, linear, or can be converted into either form. The given differential equation is \((x+1) y' = 3y + x^2 + 2x + 1\). By inspection, this is a first-order linear differential equation.

Rewrite the Equation

Rewrite the equation in the standard linear form \( y' + P(x) y = Q(x) \). Divide each term by \((x+1)\) to get \( y' - \frac{3}{x+1} y = \frac{x^2 + 2x + 1}{x+1} \). Simplify the right part to get \( y' - \frac{3}{x+1} y = x + 1 \).

Identify the Integrating Factor

The integrating factor \( \mu(x) \) is found using \( e^{\int P(x) \, dx} \). The function \( P(x) = -\frac{3}{x+1} \). Compute the integral \( \int -\frac{3}{x+1} \, dx = -3\ln|x+1| \) which simplifies to \( e^{-3\ln|x+1|} = \frac{1}{(x+1)^3} \).

Multiply through by the Integrating Factor

Multiply every term in the original equation by the integrating factor \( \frac{1}{(x+1)^3} \). Then you have: \( \frac{1}{(x+1)^3}y' - \frac{3}{(x+1)^4}y = \frac{1}{(x+1)^2} \).

Solve the Integrating Equation

Recognize that the left side is the derivative of \( \left(\frac{y}{(x+1)^3}\right) \). So it can be rewritten as \( \left(\frac{y}{(x+1)^3}\right)' = \frac{1}{(x+1)^2} \). Integrate both sides with respect to \(x\). The solution to the integral on the right is \( -\frac{1}{x+1} + C \). Thus, \( \frac{y}{(x+1)^3} = -\frac{1}{x+1} + C \).

Solve for y

Multiply through by \((x+1)^3\) to get \( y = -(x+1)^2 + C(x+1)^3 \). This represents a family of solutions.

Consider Initial Conditions

Different initial conditions will determine the value of \(C\). For example, if an initial condition specifies a particular value at a certain \(x\), solving for \(C\) based on that will give a unique solution from the family. Changes in \(C\) may affect the behavior and properties of the solution.

Key concepts

Integrating Factor

In first-order linear differential equations, the integrating factor simplifies the process, making the equation easier to solve. It's a clever multiplier that transforms a difficult equation into one that's more manageable. For an equation in the standard form:

• \(y' + P(x)y = Q(x)\)

The integrating factor \( \mu(x) \) is calculated as \( e^{\int P(x) \, dx} \).
This formula might seem intimidating, but it is just the exponential of the integral of \( P(x) \). In our exercise, \(P(x) = -\frac{3}{x+1}\). Calculating \(\int -\frac{3}{x+1} \, dx\) results in \(-3 \ln|x+1|\). Exponentiating this gives the integrating factor \( \frac{1}{(x+1)^3} \).
This helps to recast the differential equation in a form that is straightforward to integrate, easing the path to the solution.

Initial Conditions

Initial conditions provide the specific values that will help define a unique solution in a family of solutions. They determine constants of integration, ensuring that the solution is not just general but specific to a situation. Typically, an initial condition will be given in the form of a function value, say \( y(x_0) = y_0 \).
In our differential equation, after finding the general solution, \(y = -(x+1)^2 + C(x+1)^3\), we utilize initial conditions to solve for \( C \).
This means substituting the given point \((x_0, y_0)\) into the equation and solving for \( C\). Each initial condition alters the value of \( C \), resulting in a specific curve from the family of solutions, thereby tailoring the differential equation to meet specific scenarios.

Solution Behavior

The behavior of a solution of a differential equation is influenced by the values of constants and initial conditions. For the solved family of solutions, \( y = -(x+1)^2 + C(x+1)^3 \), changing \( C \) modifies the curve.

• If \( C \) is large, the term \( C(x+1)^3 \) dominates the behavior, making the solution grow quickly.
• If \( C \) is zero, the solution's behavior is solely dependent on \(-(x+1)^2\), resulting in a parabolic curve.

Different initial conditions will result in different \( C \) values, affecting how rapidly or slowly solutions rise or fall. Such analysis is critical in understanding and predicting how systems described by differential equations will behave under varying conditions.
Studying solution behavior is essential in fields like engineering and physics, where predicting outcomes based on initial settings is vital.