Consider the differential equation on the square and .

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Consider the differential equation dydx=x+12y\displaystyle \frac{dy}{dx}=x+\tfrac{1}{2}y on the square 0≤x≤40\le x\le4 and −2≤y≤2-2\le y\le2. (a) Compute dydx\frac{dy}{dx} at each of the following nine lattice points: (0,0), (0,2), (0,−2), (2,0), (2,2), (2,−2), (4,0), (4,2), (4,−2)(0,0),\,(0,2),\,(0,-2),\,(2,0),\,(2,2),\,(2,-2),\,(4,0),\,(4,2),\,(4,-2). (b) Using the slopes found in part (a), sketch the corresponding direction segments on a set of axes. Include axes labels and scale. (c) Along the line y=−xy=-x inside the square, indicate whether the solution curves are increasing, decreasing, or horizontal. Support your conclusion with appropriate slope calculations.

Related theory

Browse the problem library