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Differential equations

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1. Considering the differential equation y' = (y/x)3 :
a. Discuss existence and uniqueness of solutions.
b. Determine if exist constant solutions.
c. Determine the general integral.
d. Solve Cauchy problems y(3) = -1, y(3) = 0 and determine the maximal interval of solutions.

2) Integrate the differential equation

3) Solve the Cauchy problem:
y'' + y' = sin x
y(0) = y'(0) = 0

4. Calculate the volume of cylinder with generatrix parallel to z axis included between T domain = and the part of surface of equation z = log(xy) , projected on T.

5.
The integral ... where T is the square with vertex (0,1) , (1,0), (0,-1), (-1,0)
A. Is equal to the area of the square
B. Is ½ of the area of the square
C. Is reducible to a multiple of an integral in the first quadrant
D. Is reducible to a multiple of an integral calculated in the first quadrant under the line y=x
E. The value can be determined without the direct calculation of the integral

Solution Summary

This provides examples of a variety of differential equation problems, including Cauchy problems, integration, volume, and curvilinear abscissa.

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