Maple and Solutions of Differential EquationsGoal: Explore differential equations using a variety of representations (numerical, symbolic, and graphical) through pattern exploration assisted by appropriate technology, including the computer algebra system Maple, which is one of the course goals.Read the text and hit return in each Maple command line (the commands are in red). 1. By-hand solution of y'=y t Solve the separable differential equation y'=y t or equivalently dy/dt = y t with the initial condition y(0) = .5 Show work to solve for y, including where you use the initial condition. 2. Hit return at the following commands, which will give the general solution, and the solution with the initial condition. with(DEtools): with(Student[NumericalAnalysis]): dsolve(diff(y(t),t)=y(t)*t); dsolve({diff(y(t),t)=y(t)*t, y(0)=.5}); Maple can quickly solve DEs numerically, graphically and algebraically! 3. Comparing Maple and by-hand work.Compare your by-hand solution in #1 to Maple's solution in #2 and resolve any differences. 4. Sketch slope field tick marksAt the two points [0, 0.5] and [0.5, 0.5] solve for and sketch slope field tick marks by using the DE y'=y t. 5. Compare your slope field markings with Maple's. Compare what you drew with Maple's drawing, which should agree at the two points: DEplot(diff(y(t),t) = y(t)*t, y, t = 0 .. 1, [y(0) =.5], arrows = medium, linecolor = black); 6. Equilibrium solutions Does this DE have any equilibrium solutions? Set the original DE equation equal to 0 (not the solution) and look for constant y-value solutions, if any. 7. Apply Euler's method two times Use Euler's method, applied twice, starting at the point [0, .5] with \316\224t = .5t y y' =dy/dt 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 8. Underestimate or overestimate Using the slope field graph of the solution through [0, .5] in #5, will Euler's method be an underestimate or overestimate? 9. Euler's method in MapleGenerate graphical and numerial representations of Euler's method in Maple: Graphical: Euler(diff(y(t),t) = y(t)*t,y(0)=0.5,t=1, output=plot, numsteps=2); Numerical: Euler(diff(y(t),t) = y(t)*t,y(0)=0.5,t=1,numsteps=2); 10. Comparing Maple and by-hand work.Compare your by-hand solution in #7 to Maple's numerical solution in #9 and resolve any differences. 11. Growing or shrinking proportional to the amount present Write a separable differential equation showcasing when the rate of change is proportional to the amount present. Select a specific rate of change. Notice that multiplication in Maple is * inside the commands and you will need to modify the DE by changing =y(t)*t to your new DE in all the commands below, so the *t will have to change to a constant of proportionality: The solution with the initial condition (change =y(t)*t to your new DE) dsolve({diff(y(t),t)=y(t)*t, y(0)=.5}); Slope field (change =y(t)*t to your new DE) DEplot(diff(y(t),t) = y(t)*t, y, t = 0 .. 1, [y(0) =.5], arrows = medium, linecolor = black); Euler's method (change =y(t)*t to your new DE) Euler(diff(y(t),t) = y(t)*t,y(0)=0.5,t=1, output=plot, numsteps=2); Euler(diff(y(t),t) = y(t)*t,y(0)=0.5,t=1,numsteps=2); Question: At the point [0, .5], will Euler's method be an underestimate or overestimate? 12. Newton's Law of Colling or Heating Next write a separable differential equation representing Newton's Law of Cooling or Heating. Select some specific numbers for a negative constant of proportionality (so -k, where k is positive) and a specific ambient temperature for this DE. Thus you will have =y(t)*t below modified to -k*(y(t) - ambient temp) with a specific k and ambient temperature, i.e. all but y(t) as constants. Notice that multiplication in Maple is * inside the commands and you will need to modify the DE by changing =y(t)*t to your new DE in all the commands below: The solution with the initial condition (change =y(t)*t to your new DE) dsolve({diff(y(t),t)=y(t)*t, y(0)=.5}); Slope field (change =y(t)*t to your new DE) DEplot(diff(y(t),t) = y(t)*t, y, t = 0 .. 1, [y(0) =.5], arrows = medium, linecolor = black); Question: What is the equilibrium solution? Given the real-life context of the problem, is it stable, unstable, or neither? You can modify the initial condition to be above and below the equilibrium solution (by changing the .5 in y(0) =.5) and also increase the 1 in t = 0 .. 1 to other values. Euler's method (change =y(t)*t to your new DE) Euler(diff(y(t),t) = y(t)*t,y(0)=0.5,t=1, output=plot, numsteps=2); Euler(diff(y(t),t) = y(t)*t,y(0)=0.5,t=1,numsteps=2); Question: At the point [0, .5], will Euler's method be an underestimate or overestimate? Next go back to the class highlights page.