Slope Fields (6.1)Slope Fields (6.1)March 12th, 2013March 12th, 2013

I. General & Particular Solutions

A function y=f(x) is a solution to a differential equation if the equation is true when y and its derivatives are replaced by f(x) and its derivatives.

Ex. 1: Determine whether the function y=5lnx is a solution of the differential equation y(4)-16y=0.

You Try: Determine whether the function y=x2ex is a solution of the differential equation xy’-2y=x3ex.

The general solution of a first-order differential equation represents a family of solution curves that include a constant C. Given initial conditions, we can obtain a particular solution of the differential equation.

Ex 2: Find the particular solution of the differential equation 3x+2yy’=0 that satisfies the initial condition y=3 when x=1, given that the general solution is 3x2+2y2=C.

Ex. 3: Use integration to find a general solution of the differential equation .

dy

dx=xcosx2

You Try: Use integration to find a general solution of the differential equation .

dy

dx=5e−x/2

II. Slope FieldsSlope fields, or direction fields, can be used to analyze the solution to a differential equation that cannot be found analytically. Given a differential equation of the form y’=F(x, y) we can plug in any point (x, y) to find the slope y’ of the solution of the equation at that point. The sketch of short line segments that show the slope at each point is called a slope field.

Ex 4: Sketch a slope field for the differential equation .

dy

dx=2−y

Ex. 5: (a) Sketch the slope field for the differential equation y’=y+xy, (b) Use the slope field to sketch the solution that passes through the point (0, 4), and (c) discuss that graph of the solution as and .

x→ ∞x→ −∞