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 \(y(0)=-2\)

Short answer

The function has critical points at \( y = \pm \sqrt{\frac{4}{3}} \). Plotting shows the interval of existence excludes these, around \( t = 0 \) with \( y(0) = -2 \).

Step-by-step solution

Understand the Differential Equation

The given differential equation is \( y' = \frac{3 \sin t}{3y^2 - 4} \). This equation relates the derivative of \( y \) with respect to \( t \) to \( y \) and \( t \) itself. The initial condition provided is \( y(0) = -2 \).

Identify the Implicit Function

The differential equation is given implicitly in terms of \( y \) and \( t \). This means that \( y \) cannot be easily isolated, and we should consider it implicitly defined. We need to consider solving this equation by integrating or looking at the behavior of \( y \) given initial conditions.

Compute the Critical Points and Behavior

We need to identify the critical points where \( 3y^2 - 4 = 0 \), since these are points at which the denominator becomes zero and the function could be undefined. Solving \( 3y^2 - 4 = 0 \), we get \( y = \pm \sqrt{\frac{4}{3}} \). These points suggest potential vertical asymptotes in the behavior of the solution.

Plot Using `ezplot`

`ezplot` is a command in MATLAB designed to plot implicit equations. Plot the function using `ezplot` over a reasonable domain for both \( t \) and \( y \). This domain should include \( t = 0 \) and extend around \( y = -2 \) to see how the solution behaves and where it approaches critical values.

Estimate the Interval of Existence

From the plot generated in Step 4, analyze where the solution remains finite and continuous. This typically means noting where it avoids the critical values \( y = \pm \sqrt{\frac{4}{3}} \). The interval around the initial condition within which this holds will be the interval of existence for the solution.

Key concepts

Implicit Functions

An implicit function is a relationship where the function itself is not explicitly defined for one of the variables. In the context of ordinary differential equations, you can think of it as having a relationship between variables such as \( y \) and \( t \) without directly expressing \( y \) as a standalone expression in terms of \( t \). This contrasts with explicit functions where one variable is solely on one side of the equation.

### Understanding the EquationIn our equation \( y' = \frac{3 \sin t}{3y^2 - 4} \), \( y \) is not isolated, meaning \( y \) is implicitly defined within the equation alongside \( t \). This requires special methods for determining the behavior and solutions of the equation, often involving integration or numerical methods.

### The Role of Implicit SolutionsImplicit solutions signal scenarios where direct computation might not be straightforward, thereby needing alternative methods such as plotting or numerical analysis to understand the function's behavior fully. Especially in differential equations, they highlight the intricate relationship between \( y \) and \( t \) that isn't easily observed from the equation directly.

MATLAB Plotting

MATLAB is an incredibly powerful tool used for numerical computing, and one of its special features is the ability to plot implicit functions. The `ezplot` command in MATLAB allows you to visualize equations without needing to explicitly rearrange them.

### Using 'ezplot' CommandIn the given exercise, the `ezplot` function is used to plot the differential equation \( y' = \frac{3 \sin t}{3y^2 - 4} \). This command helps to visualize the function's behavior over a chosen domain, which might otherwise be challenging to decipher due to its implicit nature.

• Choose a relevant domain for \( t \) that adequately shows its behavior.
• Adjust the range for \( y \) to ensure you capture critical behaviors and values mentioned in the equation.

This command is particularly useful in understanding where the solution may approach asymptotic behaviors or critical points.

Interval of Existence

The interval of existence for a differential equation’s solution defines the range over the independent variable \( t \) within which the solution is valid or defined. This concept is crucial because it showcases where the function remains continuous and avoids infinite or undefined outcomes.

### Determining from PlotTo determine the interval of existence from a plot:

• Identify where the solution remains stable without discontinuities or approaching infinity.
• Avoid the critical points which in this exercise are given by \( y = \pm \sqrt{\frac{4}{3}} \).

In this particular equation, \( y \) must not reach these values to maintain a valid solution area. Analyzing the plot helps estimate these intervals visually, ensuring you're examining meaningful portions of the solution space.

Initial Conditions

In differential equations, initial conditions are pivotal as they provide the unique starting point necessary to solve the equation. The specified initial condition "pins down" one solution out of many possible ones that satisfy the differential equation.

### Role in SolutionsFor the equation \( y' = \frac{3 \sin t}{3y^2 - 4} \), we have the initial condition \( y(0) = -2 \). This means at \( t=0 \), the value of \( y \) is known and will guide the solving process.

• This condition must be factored throughout the solution process to maintain coherence.
• Helps avoid arbitrary solutions by providing a concrete starting reference point in our interval of existence.

Initial conditions are hence crucial in boundary value problems and ordinary differential equations to arrive at a specific solution graph that respects the physical or theoretical settings of the problem.