Question

Direction fields can be used to approximate the level curves of the partial differential equation \(a(x, t) u_{x}+b(x, t) u_{t}=0\). (a) Consider the rectangular region in the \(x t\)-plane described by \(-2 \leq x \leq 2\), \(0 \leq t \leq 2\). In this region, sketch the direction field for the given partial differential equation, and use the direction field to sketch estimates of the level curves of the differential equation. (b) Suppose we require that \(u(x, 0)=x,-2 \leq x \leq 2\). This constraint sets the actual elevations of the level curves constructed in part (a). In particular, note that \(u(0,0)=0\). Consider the point \((x, t)=(0,1)\) in the domain and the corresponding value of the solution, \(u(0,1)\). Based on the construction made in part (a), do you anticipate that \(u(0,1)\) will be greater than, equal to, or less than zero? \((t+1) u_{x}+u_{t}=0\)

Short answer

Question: Analyze the given partial differential equation and use the level curves to predict the value of \(u(0,1)\). The equation is \((t+1) u_{x} + u_{t} = 0\) with the initial condition \(u(x,0) = x\). Answer: Based on the level curves obtained from the direction field, we predict that the value of \(u(0,1)\) will be less than zero.

Step-by-step solution

Analyze the Partial Differential Equation to Obtain Slope

Recall that a direction field of a partial differential equation represents the slope of the tangent where the arrows' angles vary. Here, we are given the equation \((t+1) u_{x} + u_{t}=0\). Let's solve it for the slope of the tangent: Consider \(u_{t} = - (t+1) u_{x}\). Since we want the slope of the tangent, we are interested in the ratio \(u_t/u_x\). Therefore: $$\dfrac{u_{t}}{u_{x}} = - (t+1)$$

Sketch the Direction Field

For this step, take a piece of grid paper and mark off the rectangular region \(-2 \leq x \leq 2\), \(0 \leq t \leq 2\). For each point \((x, t)\) in this region, draw an arrow with slope \(u_t/u_x = - (t+1)\). You can use a ruler or another straightedge to ensure that your arrows are relatively even in length.

Sketch the Level Curves

Level curves represent the paths on which the function \(u(x,t)\) has the same value. To sketch the level curves, follow the direction field arrows but with the initial constraint that \(u(x, 0)=x\) for \(-2 \leq x \leq 2\). This means that you will start at each point \((x,0)\) along the \(x\)-axis and trace paths that follow the direction field.

Analyze \(u(0, 1)\)

Part (b) asks us to predict if the value of \(u(0, 1)\) will be greater than, equal to, or less than zero. We can do this by following the level curve that starts at \(u(0, 0) = 0\). Starting at \((0,0)\) and following the direction field curve, we can see that the curve goes into negative \(x\) values for increasing \(t\), as the slope of the tangent gets steeper with the increase in the value of \(t\). Therefore, the function \(u(x, t)\) should have a negative value at the point \((0,1)\). As a conclusion, we anticipate that \(u(0,1)\) will be less than zero.