MATH 4A Lecture Notes - Lecture 9: Slope Field
Document Summary
Today we will discuss a method of numerically approximating solutions to first order. From e/u theorem, f(t,y) is continuous in y and t. df/dy is also continuous in y and t. But we can"t find a nice formula for the unique solution. The slope field gives us a lot of qualitative equation about the solution. The slope field can help us approximate values of the solution. y(0)=-1 y-y0=(t-t0)m y=y0+(t-t0)[1/2cos(y0)+t0] y=-1+(t-0)[1/2cos(-1)+0] y1=y0+1/2f(t0,y0)=y0+1/2[1/2cos(y0)+t0]=y0+1/2[1/2cos(-1)+0]. Need equation for tangent line through (tn,yn) y-yn=(t-tn)[1/2cos(yn)+tn] y=y(n+1) t=t(n+1)=tn+h h is time step. y(n+1)=yn+h[1/2cos(yn)+tn] Example: y"=1/2y-t y(0)=2 y"-1/2y=t (u(t)=e^(-t/2)) ye^(-t/2)=2te^(-t/2)+4e^(-t/2)+c y(t)=2t+4+ce^(t/2), y(0)=2, 2=4+c, c=-2 y(t)=2t+4-2e^(t/2) (y-y0)/(t-t0)=m y-y0=(t-t0)m y=y0+(t-t0)m y(0)=2 y=2+(t-t0)(1+0)=2+t-t0. Consider the ivp for bacteria in a petrie dish from the first day of class: p. = 0. 5p+10, p(0)=60 using euler"s method with h = 1. Notice, this is exactly the same as the difference equation we got for m&m"s with death and immigra-tion.
