Question

In Exercises \(29-34\) , construct a slope field for the differential equation. In each case, copy the graph at the right and draw tiny segments through the twelve lattice points shown in the graph. Use slope analysis, not your graphing calculator. $$\frac{d y}{d x}=2 x+y$$

Short answer

The slope field for the given differential equation can be created by calculating the slopes at all given lattice points, and then graphically representing these slopes with small line segments centered at their corresponding points. This gives us a visual representation of the solutions to the differential equation.

Step-by-step solution

Understand the Equation

The given differential equation is \( \frac{d y}{d x}=2 x+y \). This equation tells us how the function \( y \) changes with respect to \( x \). For each point in the space (x, y), the derivative (slope) at that point is given by \( 2x + y \).

Calculate the Slopes

Slope fields are constructed by computing the slope of the function at several points. This is done by inserting the coordinates of the point into the differential equation. For example, at the point (1,1), the slope would be \( 2*1 + 1 = 3 \). Perform this calculation for all available lattice points.

Draw the Slope Field

Once the slopes at all given points are calculated, we graphically represent these slopes at their corresponding points. For each point, draw a small line segment centered at the point and having the calculated slope. This results in a vector field which visually represents the solution to the differential equation.