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Differential Equations and Normal Line

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Question 1.

1) Find a vector normal to the surface z + 2xy = x2 + y2 at the point (1,1,0).

2) Determine if there are separable differential equations among the following ones and explain:

a) dy/dx=sin(xy),

b) dy/dx = (xy)/(X+y)

c) dr/d(theta) = (r^2+1)cos(theta)

3) Find the general solution of the differential equation

dy/dx = (3x^2)/ (y - sin y)

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Solution Summary

This solution tells what is a separable differential equation and explains how to solve a separable differential equation.

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Differential Equations Formatting

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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.

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

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