Math 325: Differential Equations	Assignment 1		Fall 1997

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Section:

Problem 1

Verify that the functions
	y1[t_] = 1;      y2[t_] = t;
	y3[t_] = E^(-t); y4[t_] = t*E^(-t);
are solutions of the differential equation
	y''''[t] + 2y'''[t] + y''[t] == 0
and calculate their Wronskian.

Solution

Problem 2

Show that the Wronskian of Cos[t], Cos[t]^3, Cos[t]^5, and Cos[5t] is 0.
Establish this result without direct evaluation of the Wronskian.

Solution

Problem 3

Find all the 10th roots of 1024, including complex numbers. Convert these to points in the plane and plot them.

Solution

Problem 4

Find the roots of the characteristic equation and the corrseponding fundamental solutions to the differential equation
	y''''[t] + 6y'''[t] + 17y''[t] + 22y'[t] + 14y[t] == 0
Superimpose plots of the fundamental solutions.Then find and plot the particular solution that satisfies the initial conditions:
	y[0] == 1, y'[0] == 1, y''[0]==1, y'''[0] == 1
Compare your solution with the solution given by DSolve[].

Solution

Problem 5

Find the general solution to the following differential equation using the method of undetermined coefficents.
	y''''[t] + 2y'''[t] + 5y''[t] + 8y'[t] + 4y[t]==Sin[2t] + t*E^(-t)
Superimpose plots of the particular solutions satisfying
	y[0] == y0, y'[0] == 0, y''[0] == 0, y'''[0] == 0
for several values of y0. Explain why all these solutions seem to pass through the t-axis at the same points and at regular intervals for large values of t. Try to determine the length of this common interval.

Solution

Problem 6

Find approximate values of the solution of the initial value problem
	y'[t] == (t^2 - y[t]^2) Sin[y[t]], y[0] = 1
using the Euler method with h = 0.1, h = 0.05, and h = 0.025. Compare these results with the output of NDSolve[] by superimposing plots of the data.

Solution

Hints: Define a function that does one step of the Euler method, e.g.,
	EMStep[{t_,y_}] = N[{t+h,y+h*f[t,y]}];
Then use NestList[EMStep, {t0,y0}, n] to apply EMStep[]
successively to its own output, starting at the point {t0,y0} and continuing for n steps.
To plot the results use ListPlot[...,PlotJoined->True].  The output of NDSolve[] is a list (in this case a list of one!) of substitution rules of the form
{y[t]->InterpolatingFunction[...][t]}. To plot this, do something like
sol = NDSolve[...]; Plot[y[t]/.First[sol],{t,a,b}]. You can combine plots with Show[].

Problem 7

Find approximate values of the solution of the initial value problem
	y'[t] == (t^2 - y[t]^2) Sin[y[t]], y[0] = 1
using the Runge-Kutta method with h = 0.1, h = 0.05, and h = 0.025. Compare these results with the output of NDSolve[]  (from Problem 6) by superimposing plots of the data.

Solution

Hints: Use the same ideas as in Problem 6. For the Runge-Kutta step, use the formula on p.437 of the textbook. Several lines of calculation can be done at once by enclosing them in parentheses, e.g.,
	 RKStep[{t_,y_}] = ( ...; ...; ...; N[{...,...}]);