Section 1. Entering, Solving, Plotting
Unevaluated derivatives are used to enter a differential equation, dsolve solves it. The phi function can be used to plot families of solutions satisfying specified initial conditions.
0. Pull down the Help menu, choose New Users/Full Tour, then click on the item eight: Differential Equations and work through the ordinary differential equation examples in that worksheet.
Open a new Maple worksheet and do the following
1. Note that the New Users worksheet on differential equations uses the D operator to enter a differential equation. Enter the differential equation y' + y = sin(t) using the D operator. Name it DE1. Obtain the general solution and the solution satisfying y(0) = 0.
2. Continuing 1. Enter the same differential equation using the diff procedure. Name it DE2. Obtain the general solution and the solution satisfying y(0) = 0. Compare the solutions to the ones found in Exercise 1.
We recommend that you enter differential equations using the diff procedure.
3. Obtain the general solution to the equation y' = t /cos(y) by entering the equation with the name DE and using
dsolve( DE );
Now solve the equation using
dsolve( DE, implicit );
4. Continuing 3. Obtain the solution to DE satisfying y(0) = 1. Do it twice, once using
dsolve( {DE,y(0)=1} );
and again using
dsolve( {DE,y(0)=1}, implicit );
5. Obtain the general solution to the equation
and plot the solutions corresponding to _C1 = -2, -1, 0, 1, 2.
Hint. Experiment with the horizontal plot range until you get nice picture displaying all 5 curves near t = 0. Use the optional equation axes=framed.
6. Continuing 5. Obtain the solution to DE satisfying y(1) = 1. Plot it over a reasonable interval containing t = 1.
Use unapply to convert the solution into a function f. Use f to generate a sequence of solution values for t = 0, 0.25, 0.5, 1.25, 1.50, 2.0.
7.* Enter the differential equation y' + y = cos(t) with the name DE. Obtain the solution satisfying the generic initial condition y(t0) = y0. Name it soln.
a. Use the entry eval( soln, t=t0) to verify that soln satisfies the initial condition.
b. Use unapply to make rhs(soln) into a function of t, t0, and y0 named phi (see page 37 in Part III, Section 1 of the manual).
c. Use phi to plot some solutions starting at points evenly spaced on the y axis.
d. Use phi to plot some solutions starting at points evenly spaced on the t axis.
e. Use phi to plot the solutions starting at points evenly spaced around the unit circle in the style of the two plots on page 35 of the manual. That is, one picture runs time forward, another runs time backward.
8. Continuing 7. Examine the formula for the phi function and describe the behavior of all solutions as t approaches infinity.
9. Look before you leap.
Consider the following differential equation.
a. What is the formula for the function f such that this equation is equivalent to
b. Based upon the statement of the Unique Solution Theorem at what points do you expect that solutions will fail to exist and/or fail to be unique?
c. Obtain the phi function for this differential equation and use it to obtain the solutions described in parts c, d, and e of Exercise 7.
10. Obtain an informative picture of the family of solutions to
over the interval t < -1. Hint. Use the phi function. (See page 38 in the manual.)