Question

The solutions are defined implicitly. Find the solution and plot it using ezplot in a region which displays the most important aspects of the solution. On the basis of your graph estimate the interval of existence. \(y^{\prime}=3 \sin t /\left(3 y^{2}-4\right)\) with \(y(0)=0\)

Short answer

The solution exists around \( t = 0 \), constrained by \( y = \pm\sqrt{4/3} \).

Step-by-step solution

Understand the Differential Equation

The differential equation given is \( y' = \frac{3 \sin t}{3y^2 - 4} \) with the initial condition \( y(0) = 0 \). This is a first-order, nonlinear differential equation. Our task is to analyze the behavior of this equation graphically to find the region where the solution is valid.

Find Critical Points of the Denominator

To ensure that the function is well-defined, the denominator of \( y' = \frac{3 \sin t}{3y^2 - 4} \) must not be zero. Solve \( 3y^2 - 4 = 0 \) to find the critical points. This simplifies to \( y^2 = \frac{4}{3} \), giving \( y = \pm \sqrt{\frac{4}{3}} \). We must ensure \( y(t) \) stays within these bounds to avoid division by zero.

Solve the Initial Condition

Given \( y(0) = 0 \), substitute this into the equation \( y' = \frac{3 \sin t}{3y^2 - 4} \) to determine the behavior at \( t = 0 \). Check if \( y(t) = 0 \) satisfies this equation at the initial condition, which it does since \( \sin(0) = 0 \).

Analyze the Behavior of the Solution

To understand the solution further, consider the nature of \( \sin t \) which oscillates between -1 and 1. The solution \( y(t) \) will be impacted by this oscillatory behavior, taking care not to allow \( 3y^2 - 4 \) to reach zero.

Plot Using EZPlot for Visual Analysis

Use a symbolic plotting tool such as `ezplot` in MATLAB or a similar tool in Python to graph \( y' = \frac{3 \sin t}{3y^2 - 4} \). Set a range for both \( t \) and \( y \) values that cover the initial condition and reveal the behavior close to the critical points. Observe where the solution appears to remain stable and bounded by the critical points.

Estimate Interval of Existence

From the plot, determine the interval of \( t \) over which the solution exists without \( 3y^2 - 4 \) equating to zero, ensuring you consider both positive and negative \( t \) directions from \( t = 0 \). Typically, this interval will shrink in towards zero as \( \sin t \) restricts \( y \) more closely to zero.

Key concepts

Plotting with EZPlot

Plotting with tools like EZPlot can greatly help in visualizing implicit solutions to differential equations. Here, the first step is translating the differential equation into a visual format. When given an equation like \( y' = \frac{3 \sin t}{3y^2 - 4} \), EZPlot assists in simplifying this complex expression.
The function `ezplot` is primarily used in MATLAB as a straightforward way to plot implicit functions by specifying the function and the domain. The beauty of EZPlot is that it can handle expressions that do not easily decompose into simple x and y equations.
Key benefits of using EZPlot include:

• Automatic domain selection that emphasizes significant features of the graph.
• Easy input of simple syntax, which handles equation-only inputs.
• Flexibility in exploring different aspects of the solution graphically.

When using `ezplot`, pay attention to the boundaries identified earlier in the calculus process, ensuring that problematic divisions by zero and undefined ranges are avoided in your plots.

Critical Points in Calculus

Critical points play a central role in assessing any differential equation's behavior. In the given problem, identifying where the denominator \(3y^2 - 4\) becomes zero is critical because it determines the points at which the function becomes undefined.
To find these critical points, solve the equation \(3y^2 - 4 = 0\).

• Simplifying gives \(y^2 = \frac{4}{3}\).
• This yields critical points at \(y = \pm \sqrt{\frac{4}{3}}\).

These values are crucial when ensuring the solution does not diverge, as they dictate where the function might experience undefined behavior due to division by zero.
Staying within these bounds during analysis ensures mathematical continuity and avoids places where the function might "blow up." By understanding these critical points, one can determine where solutions might exist or fail and allows plotting to reflect realistic behavior.

Interval of Existence Analysis

The interval of existence of a solution to a differential equation determines the range over which the solution remains valid and well-defined.
In our equation, this involves analyzing where the denominator \(3y^2 - 4\) is non-zero since zero would make the equation undefined. The critical points derived previously guide this interval assessment.
To determine the intervals:

• Ascertain where \(y\) remains within the bounds \(-\sqrt{\frac{4}{3}} < y < \sqrt{\frac{4}{3}}\).
• Use these to extrapolate the possible \(t\)-intervals based on the initial condition \(y(0) = 0\).

The behavior of \(\sin t\) introduces further constraints, as it oscillates between -1 and 1. These oscillations mean that indeed the solution is more closely confined by the critical points.
Ultimately, plotting helps estimate these intervals visually, observing over which range the solution holds consistent without reaching undefined values.

Nonlinear Differential Equations

Nonlinear differential equations, such as the one at hand, describe systems with complex interactions where the rate of change of the variable depends nonlinearly on the variable itself. Unlike linear systems, these equations often cannot be solved explicitly with simple functions.
In this equation, \(y' = \frac{3 \sin t}{3y^2 - 4}\), nonlinearity is evident in the quadratic term \(3y^2\) within the denominator. This nonlinearity often leads to more intricate behavior, such as:

• Sensitivity to initial conditions, affecting the overall solution.
• Possible multiple solutions or behaviors due to inherent equation complexity.
• Inability to find a closed-form solution, relying on numerical methods or graphical analysis instead.

Analyzing nonlinear differential equations usually requires careful consideration of initial conditions and stability analysis. Often leverage numerical solvers or simulations, alongside graphical tools, can provide insightful perspectives into how the solutions behave across different domains.