Question

Solve the equation explicitly. Also, plot a direction field and some integral curves on the indicated rectangular region. $$ x y y^{\prime}=x^{2}+2 y^{2} ; \quad\\{-4 \leq x \leq 4,-4 \leq y \leq 4\\} $$

Short answer

Question: Analyze and solve the first-order ordinary differential equation \(xyy' = x^2 + 2y^2\). Additionally, plot the direction field and some integral curves on the rectangular region \(-4 \leq x \leq 4, -4 \leq y \leq 4\). Answer: The given ODE, after being rewritten as \(\frac{dy}{dx} = \frac{x^2 + 2y^2}{xy}\), is found to be separable. However, we cannot find an explicit solution for \(y(x)\) in terms of elementary functions. To analyze the ODE, we can plot the direction field on the given rectangular region and use numerical methods to approximate the solution, which will give us integral curves that represent the behavior of the solution without needing an explicit formula for \(y(x)\).

Step-by-step solution

Check the separability of the ODE

First, let's rewrite the given ODE as: $$ \frac{dy}{dx} = \frac{x^2 + 2y^2}{xy} $$ We can see that the function is separable because it can be written as $$ \frac{dy}{dx} = \frac{1}{y} \cdot (x + 2\frac{y^2}{x}) $$

Integrate both sides

Now, let's integrate both sides of the equation: $$ \int y \, dy = \int (x + 2\frac{y^2}{x}) \, dx $$ Integration results in: $$ \frac{1}{2}y^2 = \frac{1}{2}x^2 + \frac{2}{3}y^3 + C $$ Where \(C\) is the integration constant.

Solve for y(x)

Now, we want to solve for \(y(x)\). The equation is: $$ \frac{1}{2}y^2 = \frac{1}{2}x^2 + \frac{2}{3}y^3 + C $$ Unfortunately, it's not possible to obtain an explicit solution in terms of elementary functions for \(y(x)\) as the equation is nonlinear. However, we can still plot the direction field and integral curves without finding an explicit solution.

Plot direction field

To plot the direction field, we consider the slopes of the tangent lines for the integral curves given by: $$ \frac{dy}{dx} = \frac{x^2 + 2y^2}{xy} $$ Then, we plot the direction field on the rectangular region \(-4 \leq x \leq 4, -4 \leq y \leq 4\).

Plot integral curves

Although we could not find an explicit solution for \(y(x)\) in Step 3, we can still plot the integral curves using a numerical method such as Euler's method or Runge-Kutta method to approximate the solution at specific points within our rectangular region. By doing so, we will obtain several integral curves that give a graphical representation of the solutions to our ODE for different initial conditions. By following these steps, we were able to appreciate the qualitative behavior of our ODE solution without an explicit formula for \(y(x)\).

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve functions and their derivatives. They play a crucial role in modeling how things change over time or space. In these equations, the unknown function is usually dependent on a single variable. For example, in the equation given in the problem, the variable is \(x\), and the function \(y\) depends on \(x\). The derivative of \(y\) with respect to \(x\), noted as \(\frac{dy}{dx}\), gives us information on how \(y\) changes as \(x\) changes. ODEs appear in various fields such as physics, engineering, and biology, and they can describe anything from simple motion to complex dynamic systems.
Understanding ODEs is foundational for grasping concepts in calculus and solving real-world problems that involve dynamic changes.

Separable Equations

Separable Equations represent a specific type of ODE that can be rewritten so that the variables can be separated on each side of the equation. In our problem, the ODE can be rearranged so that all terms involving \(y\) and \(dy\) are on one side, and terms involving \(x\) and \(dx\) are on the other. This separation allows us to integrate both sides independently:
\[ \int y \, dy = \int (x + 2\frac{y^2}{x}) \, dx \]
After integrating, we compare and find a relationship involving an integration constant \(C\). This constant often represents initial conditions or specific cases within a range of general solutions. Separable equations are especially useful because they enable us to solve more complex ODEs that can be reduced to a simpler form.

Direction Fields

Direction Fields, also known as slope fields, provide a visual representation of a differential equation without needing to find the exact solution. They display small line segments or arrows at various points in a plane, indicating the slope of the solutions at those points. For the equation given, plotting the direction field involves evaluating \(\frac{dy}{dx}\) at several points within the rectangular region \(-4 \leq x \leq 4, -4 \leq y \leq 4\).
This graphical approach helps us intuitively understand the behavior and direction of the solution curves, even if we can't solve the equation directly. By observing the alignment and flow of the arrows, we can predict how solutions will behave near those points.

Numerical Methods

Numerical Methods offer ways to approximate solutions to differential equations when analytical solutions are difficult or impossible to obtain. These techniques provide approximate values at discrete points and can be highly accurate. Two common methods are Euler's method and the Runge-Kutta method.

• Euler's Method: A straightforward approach that uses tangent line approximations. It takes small steps along the slope, offering a piecewise linear approximation of the curve.
• Runge-Kutta Method: Often more accurate than Euler's method. It considers the slope at multiple points within each interval to produce a better approximation of the curve.

In the context of our problem, these methods allow us to sketch approximations of integral curves, offering a glimpse into potential solution trends across our defined region. They are invaluable tools in scenarios where exact solutions are not feasible.