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integrals, vector fields, and differential equations

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1) To find the general integral of the differential equation, discuss existence et uniqueness and find the particular integral that passes for the point (1;5/2)

2) Considering the linear equation of the 2nd order
z'' - (2/x) z' + (2/ x2)z = 10 / x2 , x > 0 .

If we know that the functions:
y1(x) = 2x+5 , y2(x) = x2 +5 , y3(x) = x2 +2x+5
are solutions of the equation, write the general integral, justifying the procedure.

3) To find the general integral of the differential equation

4) To calculate = (see attached file) (sin (3y) - x y2 + 2 ) dx dy
Where E is an ellipse with center in the origin and semi-axes with length 1 and 2, that has the Cartesian axes as axis of symmetry, avoiding not necessary calculation.

5) To draw the integration's domain and to exchange the order of integration of the following double integral, calculating the value:

6) Calculate the following double integral where

7. To calculate the curve integral of the function: f(x,y) = x2 - 2y2 along the line OAB, where O(0,0), A(-2,1) B(1,1).

8) To consider the vector field V(x, y) = (V1, V2) = (( y2 - 1/x, 2xy + 1/y).
To verify that the field is conservative in the set A = {(x, y ) R2 : x > 0, y > 0 }.
To determine the potential function that is cancel out in (1, 1).

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Solution Summary

This is a series of problems regarding curve integrals, double integrals, vector fields, and differential equations.

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