Question

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves. \(2 y^{\prime}=3(y-2)^{1 / 3}, \quad y=3\) when \(x=1\)

Short answer

The particular solution is \( y = x^{3/2} + 2 \).

Step-by-step solution

- Separate Variables

Start by rewriting the given differential equation in a separable form. The equation given is \[ 2 y' = 3(y - 2)^{1/3} \]. Divide both sides by \( (y - 2)^{1/3} \) and multiply both sides by dx: \[ \frac{2 dy}{(y - 2)^{1/3}} = 3 dx \].

- Integrate Both Sides

Integrate both sides of the equation. \[ \frac{2}{(y - 2)^{1/3}} dy = 3 dx \]. Perform the integration: \[ \frac{2}{-1/3 + 1} (y - 2)^{2/3} = 3x + C \]. Simplify the integration result: \[ 3 (y - 2)^{2/3} = 3x + C \].

- Solve for Arbitrary Constant

Use the given boundary condition \( y = 3 \) when \( x = 1 \) to solve for the constant C. Substitute \( x = 1 \) and \( y = 3 \): \[ 3 (3 - 2)^{2/3} = 3 \times 1 + C \]. Simplify: \[ 3 (1) = 3 + C \]. Therefore, the constant is: \[ C = 0 \].

- Write the Particular Solution

Substitute the value of C back into the integrated equation to get the particular solution: \[ 3 (y - 2)^{2/3} = 3x \]. Solving for y, \[ (y - 2)^{2/3} = x \]. Then raising both sides to the 3/2 power, \[ y - 2 = x^{3/2} \], finally \[ y = x^{3/2} + 2 \].

- Plot the Slope Field and Solution Curves

To plot the slope field and some solution curves, use a graphing tool or software such as Python (with libraries like matplotlib) or MATLAB. The differential equation \( 2 y' = 3 (y - 2)^{1/3} \) can be plotted along with the solution curve \( y = x^{3/2} + 2 \) for a visual representation.

Key concepts

Separation of Variables

To solve a differential equation, we often start by rearranging it into a form where we can separate the variables—hence the name 'separation of variables'. This method works well for first-order ordinary differential equations where one side of the equation involves only the independent variable and its differential (often dx), while the other side involves only the dependent variable and its differential (dy). In our problem, we started with the equation \[ 2 y' = 3(y - 2)^{1/3} \] and rearranged it to isolate all terms involving y and dy on one side and all terms involving x and dx on the other: \[ \frac{2 dy}{(y - 2)^{1/3}} = 3 dx. \] Once rearranged, each side can be integrated independently.

Boundary Conditions

Boundary conditions are specific values given for the dependent variable, usually at a specific point for the independent variable. These conditions help us find a particular solution to the differential equation, as they allow us to determine any arbitrary constants introduced during integration. In our example, the boundary condition is given as \(y = 3\) when \(x = 1\). This means that when x equals 1, y must equal 3. By substituting these values back into the equation after integration, we solve for the constant C. In step 3 of our solution, using the boundary condition helped us find the value of C as 0.

Integral Calculus

Integral calculus involves the process of integration, which is essentially the reverse of differentiation. When applying separation of variables, both sides of the rearranged equation are integrated separately. For our example, after separating the variables, we integrated each side as follows: \[ \int \frac{2}{(y - 2)^{1/3}} dy = \int 3 dx. \] The integral of the left side with respect to y involves applying the power rule for integration, and similarly, the right side is integrated with respect to x to yield: \[ 3 (y - 2)^{2/3} = 3x + C. \] This integration step often introduces an unknown constant C, which we later determine using the boundary conditions.

Slope Field

A slope field, also known as a direction field, graphically represents the solutions of a first-order differential equation without actually solving the equation. Each point `(x, y)` in the field has a small line segment or slope that indicates the slope of the solution curve passing through that point. For the differential equation \(2 y' = 3 (y - 2)^{1/3}\), we compute the slope at numerous points and plot these to form the slope field. Tools like graphing calculators, Python's matplotlib library, or MATLAB are excellent for generating these fields. The slope field provides a visual understanding and helps in approximating the behavior of the solution curves.

Solution Curves

Solution curves are the actual graphs of the solutions to a differential equation, plotted within the slope field. They follow the direction of the slopes indicated in the field. In our example, after separating the variables and integrating, we derived the particular solution: \(y = x^{3/2} + 2\). This solution curve is then plotted within the slope field to visualize how the solution behaves over a range of x values. Plotting several solution curves for different initial conditions can show how solutions dynamically evolve based on slight changes in initial values, offering deeper insight into the nature of differential equations.