Question

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \(y(t=0)=-10\) to \(y(t=0)=10\) increasing by \(2 .\) Is there some critical point where the behavior of the solution begins to change?[T] \(x y^{\prime}=y\)

Short answer

The critical point, where the solution behavior changes, is at \(x = 0\).

Step-by-step solution

Separate Variables

The given differential equation is \( x y' = y \). Start by separating the variables to express the differential equation in a form where you can integrate both sides. Divide both sides by \( y \) and multiply by \( dt \) to get: \( \frac{dy}{y} = \frac{1}{x} dx \).

Integrate Both Sides

Integrate both sides of the equation. The integration of \( \frac{dy}{y} \) yields \( \ln|y| + C_1 \) and the integration of \( \frac{1}{x} dx \) yields \( \ln|x| + C_2 \). Equating gives: \( \ln|y| = \ln|x| + C \).

Solve for y

Exponentiate both sides to eliminate the natural logarithm: \(|y| = e^{C} |x|\). Since \(e^{C}\) is a constant, we can denote it as \(C'\) (where \(C' = e^C\)). Thus, the general solution is \( y = C' x \).

Apply Initial Conditions to Determine Specific Solutions

Substitute the initial conditions, \(y(0) = C' \cdot 0 = -10, -8, ..., 10\). Notice that at \(x=0\), \(y=0\) regardless of \(C'\), suggesting a horizontal line at zero for all solutions, which indicates critical behavior at this point.

Analyze Critical Points

The critical point where the solution changes behavior is at \(x=0\). At this point, the solution transitions from increasing to decreasing (or vice versa) as \(y\) moves from a positive to a negative value through zero.

Key concepts

Separation of Variables

Separation of Variables is a crucial method for solving simple differential equations. It involves rearranging an equation to isolate two variables on opposite sides. This allows us to perform integration on each side separately. For instance, given the differential equation \( x y' = y \), we aim to express each variable in terms of its own differential:

• Divide both sides by \( y \) to isolate terms with \( y \) on the left: \( \frac{dy}{y} \).
• Divide by \( x \) (or multiply by \( dx \)): \( \frac{1}{x} dx \).

This separation helps simplify integration, leading us to the solution more easily.Integrating each side will allow us to find a general relationship between \( x \) and \( y \). Keeping variables separated until they are integrated is the key aspect of this strategy.

Initial Conditions

Applying initial conditions in differential equations is essential for obtaining a specific solution from a general solution. In our case, we have the general solution derived from the differential equation, \( y = C' x \). Initial conditions are values provided, such as \( y(t=0)=-10 \), that help find specific solutions:

• Set \( x = 0 \) in \( y = C' x \), stabilizing \( y \) at initial values from \(-10\) to \(10\).
• Substitute each condition to determine distinct \( C' \) values and observe how solutions differ by these constants.

Such initial conditions narrow the general solution down to particular cases, demonstrating how mathematical models adapt to given scenarios. It highlights that solutions from these equations are highly context-dependent.

Critical Points

Critical points are particular values in the solution where behavior begins to change significantly. Here, we note a unique critical point at \( x=0 \). At this point:

• The solution \( y = C'x \) reduces to \( y = 0 \), whatever the constant \( C' \).
• Initial conditions also show horizontal lines for \( y = 0 \) around this value, where \( y \) transitions from positive to zero to negative.

This change signifies a crucial transition phase and potential instability in the system described by the differential equation. Critical points often hold valuable information on how small changes in input values can affect the output behavior of a system. They are significant when examining stability in dynamical systems.

General Solution

The general solution to a differential equation encodes all possible solutions that meet the initial equation. For example, after separating variables and integrating, we found the general solution to be \( y = C' x \). Here, \( C' \) symbolizes a constant encompassing every possible solution configuration:

• Each specific value of \( C' \) represents a different line passing through the origin, illustrating the family of solutions.
• Understanding this general pattern helps in identifying how different starting conditions (initial conditions) affect possible outcomes.

The power of the general solution lies in its flexibility; it paints a comprehensive picture of solution behavior before applying initial conditions for particular cases.