Updated 2013 2.2-10: Solve x'= -x^2 + 7x - 10 ======= Factor the quadratic as -(x^2-7x+10) = -(x-5)(x-2). Substitute P=x-5 or P=x-2 to make it into a Verhulst DE. For P=x-2 we would get Verhulst DE P' = -(x - 2 -3)(x-2) = -(P-3)(P) = (3-P)P Then P'=(a-bP)P with a=3 and b=1. Plug a,b into the equation a P_0 P(t) = ----------------------- bP_0 + (a-b P_0)exp(at) to find P(t) and then back-substitute to obtain x(t) = 2 + P(t) = something like the BOB answer. The twist: P_0 = x_0 -2, which requires even more algebra, finally matching the book's answer. MAPLE ANSWER CHECK # x'= -x^2 + 7x - 10 f:=(t,x)->-x^2+7*x-10; de:=diff(x(t),t)=f(t,x(t)); # Symbol x(t), not x! sol:=dsolve(de,x(t)); # answer x(t) = (5*exp(3*t)*_C1-2)/(-1+exp(3*t)*_C1) # Symbol _C1 is an arbitrary constant, call it c. Then # x(t) = (5*c*exp(3*t)-2)/(c*exp(3*t)-1) 2.2-14: Given x' = x(x^2 - 4) ====== Make a phase line diagram and a phase diagram Add labels: spout, funnel, node, stable, unstable Don't use a computer. Method: Step 1. Find the equilibria x=0, x=2, x= -2 Step 2. Add them to the phase line diagram Step 3. Evaluate x' = x(x^2 - 4) at t=0, x=x_0 where x_0 is located between equilibria. Example: for 0 < x_0 < 2, choose x_0 = 1 (a point between) Then | x' = x(x^2 - 4) | = 1(1^2 - 4) = negative | x = 1 Step 4. Add the PLUS and MINUS signs to the phase line diagram. Step 5. Draw the phase diagram. First, draw the equilibria, they are horizontal lines x = 0, x = -2, x = 2 Second, draw threaded edge-to-edge curves, based on the PLUS and MINUS signs. Draw an increasing threaded curve for PLUS. Draw a decreasing threaded curve for MINUS. Step 6. Identify spout, funnel and node from geometry. Then add STABLE to the funnel and UNSTABLE to the others. Put all labels onto the phase diagram. Reminder: Don't use a computer! This problem is a template for Midterm 1 problem 5. 2.2-18: Given x' = x^3(x^2 - 4) Make a phase line diagram and a phase diagram Add labels: spout, funnel, node, stable, unstable Don't use a computer. Method: Step 1. Find the equilibria x=0, x=2, x= -2 Step 2. Add them to the phase line diagram Step 3. Evaluate x' = x^3(x^2 - 4) at t=0, x=x_0 where x_0 is located between equilibria. Example: For 0 < x_0 < 2, choose x_0 = 1 (a point between) Then | x' = x^3(x^2 - 4) | = 1(1^2 - 4) = negative | x = 1 Step 4. Add the PLUS and MINUS signs to the phase line diagram. Step 5. Draw the phase diagram. First, draw the equilibria, they are horizontal lines x = 0, x = -2, x = 2 Second, draw threaded edge-to-edge curves, based on the PLUS and MINUS signs. Draw an increasing threaded curve for PLUS. Draw a decreasing threaded curve for MINUS. Step 6. Identify spout, funnel and node from geometry. Then add STABLE to the funnel and UNSTABLE to the others. Put all labels onto the phase diagram. Reminder: Don't use a computer! This problem is a template for Midterm 1 problem 5.