MATH 4A Lecture Notes - Lecture 4: Slope Field, Isocline, Implicit Function
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Definition the slope field or direction field for the differential equation y. =f(t,y) is the field of dashed lines in the ty-plane where the dash through a specific point (t. 0 y"=t-y. as long as t-y=0, t=y, y"=0. (isocline) Definition: isocline lines where y" is constant. t-y=1, y=t-1, y"=1 t-y=-1, y=t+1, y"=-1 t-y=2, y=t-2, y"=2. We can draw the slope field of the differential equation, and the slope field tells us the solution to the differential equation. Definition: an equilibrium solution to a differential equation is a constant solution. The constant function y(t) 3 is an equilibrium solution to the equation y. 1. check that y=3 is equilibrium solution to y"=3-y. 2. check that y(t)=t-1 is a solution to y"=t-y. = 9 implicitly defines the function y = sqrt(9-x^2) We say that the circle y because the graph of this function gives part of the circle of radius 3.
