Applications of exponential growth and decay



Solving first order differential equationsExample 1Solve: ?y?x=3x given that when x = 1, y = 2Example 1 solution - indefinite integral techniquedydx=3x1.dy=3x.dx1.dy=3x.dxy=32x2+cSubstitute x = 1, y = 22=32×12+c2=32+cc=12y=32x2+12Example 1 slope fieldExample 2Solve ?y?x=y2, given that when x = -3, y = 1.Example 2 solution - indefinite integral techniquedydx=y2dxdy=1y21.dx=1y2.dy1.dx=1y2dyx=-1y+cSubstitute x = -3, y = 1. (or make y the subject at this stage)-3=-11+cc=-2x=-1y-2x+2=-1y1x+2=-yy=-1x+2Example 2 solution - definite integral techniquedydx=y2dxdy=1y21.dx=1y2.dy-3x1 dx=1y1y2 dyx-3x=-1y1yx--3=-1y--11x+3=-1y+1x+2=-1yy=-1x+2Example 2 slope field Example 3Solve ?y?x=xy given that when x = 3, y = 2Example 3 solution – indefinite integral techniquedydx=xy1ydy=x dx1ydy=x dxln|y|=x22+cSubstitute x = 3, y = 2ln|2|=322+cln2=92+cc=ln2-92ln|y|=x22+ln2-92ln|y|-ln2=x22-92ln|y|2=x2-92|y|2=ex2-92y=2ex2-92y=±2ex2-92 Determine the function by substituting the values x=3 and y=2 into the positive and negative functions and testing which holds true.∴y=2ex2-92 as the point (3,2) lies on this curve.Example 3 slope field ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download