Problem 21 - Advanced Engineering Mathematics

Question

(a) A simple model for the shape of a tsunami is given by $$ \frac{d W}{d x}=W \sqrt{4-2 W}, $$ where \(W(x)>0\) is the height of the wave expressed as a function of its position relative to a point offshore. By inspection, find all constant solutions of the \(\mathrm{DE}\). (b) Solve the differential equation in part (a). A CAS may be useful for integration. (c) Use a graphing utility to obtain the graphs of all solutions that satisfy the initial condition \(W(0)=2\).

Step-by-Step Solution

Answer

Constant solutions are \( W = 0 \) and \( W = 2 \). The DE is separable; solve using integration. Graph to verify with \( W(0)=2 \).

Step 1: Finding Constant Solutions

To find constant solutions, consider solutions of the form \( W(x) = C \) where \( C \) is a constant. Substituting into the differential equation gives \( \frac{dW}{dx} = 0 \), resulting in \( C \sqrt{4 - 2C} = 0 \). This occurs when \( C = 0 \) or \( \sqrt{4 - 2C} = 0 \), the latter yielding \( C = 2 \). Hence, constant solutions are \( W = 0 \) and \( W = 2 \).

Step 2: Separate the Variables

Choose \( W eq 0 \). To solve \( \frac{dW}{dx} = W \sqrt{4 - 2W} \), first separate variables: \[ \frac{dW}{W} = \sqrt{4 - 2W} \, dx. \] Integrating both sides requires addressing the right side.

Step 3: Integration

The left side integrates directly to \( \ln |W| + C_1 \). For the right, substitute \( u = 4 - 2W \) yielding \( du = -2 \, dW \), thus \[ \int W \sqrt{4 - 2W} \, dx \rightarrow -\frac{1}{2} \int \sqrt{u} \, du = -\frac{1}{3} u^{3/2} + C_2, \] leading to \[ \ln |W| = -\frac{1}{3} (4 - 2W)^{3/2} + C. \]

Step 4: Solve for W(x)

Exponentiate to obtain \( W(x) = e^{-\frac{1}{3} (4 - 2W)^{3/2} + C} \). Solve for \( W \) to yield \( (4 - 2W)^{3/2} = -3 \ln |W| + k \), leading to a transcendental equation. Simplify further as needed to reveal solutions explicitly, using numerical methods or graphing if required.

Step 5: Graphing the Solution with Initial Condition

With the initial condition \( W(0) = 2 \), plug into the solution obtained or graph \( W(x) \) to verify the condition. Utilize a graphing utility to depict the behavior that passes through the initial point \( (0,2) \). Analyze its stability near the constant solutions found earlier: \( W = 2 \).

Key Concepts

Constant Solutions

In differential equations, constant solutions occur when the derivative of a function is zero everywhere, indicating that the function does not change over the domain. For the tsunami model equation given as \( \frac{d W}{d x} = W \sqrt{4-2 W} \), we consider solutions where the height of the wave \( W(x) \) is constant at every point. This means we substitute \( W(x) = C \), a constant, into the differential equation. Substitute to find:

• \( \frac{dW}{dx} = 0 \) implies \( C \sqrt{4 - 2C} = 0 \).
• The equation \( C \sqrt{4 - 2C} = 0 \) gives solutions when either \( C = 0 \) or \( \sqrt{4 - 2C} = 0 \).
• The latter implies \( 4 - 2C = 0 \) or \( C = 2 \).

This reveals two constant solutions: \( W = 0 \) and \( W = 2 \). These represent equilibria or steady-state wave heights under specific conditions.

Separation of Variables

The technique of separation of variables is a powerful method used to solve simple differential equations, especially when variables can be algebraically separated. Given our tsunami wave equation \( \frac{dW}{dx} = W \sqrt{4 - 2W} \), we aim to separate the variables \( W \) and \( x \) on opposite sides of the equation:

• We rewrite as \( \frac{dW}{W} = \sqrt{4 - 2W} \, dx \).
• This allows us to integrate both sides independently, handling \( W \) and \( x \) separately.

By separating them, we prepare the equation for integration, an essential step to find the general solution of a differential equation. This method is crucial, especially when the equation is nonlinear, as in this case.

Integration in Differential Equations

Once variables are separated, the next step is integration to solve the differential equation. Each side of the separated equation \( \frac{dW}{W} = \sqrt{4 - 2W} \, dx \) is integrated separately:

• The left side integrates to yield: \( \ln |W| + C_1 \).
• For the right side, substitute \( u = 4 - 2W \) making \( du = -2 \, dW \).
• This transforms the integral to \( -\frac{1}{2} \int \sqrt{u} \, du \), resulting in \( -\frac{1}{3} u^{3/2} + C_2 \).
• Substitute back to assemble the complete relation: \( \ln |W| = -\frac{1}{3} (4 - 2W)^{3/2} + C \).

Integrating is key to unlocking the behavior of the tsunami model by providing a functional relationship between the wave's height \( W \) and its position \( x \).

Graphing Solutions

Graphing solutions is an essential methodology to visually comprehend the behavior of differential equations. After solving our tsunami wave equation, you might not obtain an explicit equation for \( W(x) \), or it may be complex. Graphing helps verify the solution matches initial conditions and visualizes behavior:

• Use the initial condition \( W(0) = 2 \) to determine curves that pass through this point.
• Plot the wave height \( W(x) \) using graphing software or a tool that can handle implicit functions.
• Observe how the graph compares to the constant solutions \( W = 0 \) and \( W = 2 \), focusing on their stability and role as equilibrium points.

Graphically understanding differential equations offers intuition on dynamics and changes, especially when analytical solutions are challenging.