integrals, vector fields, and differential equations

Please see attached

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 =  (sin (3y) - x y2 + 2 ) dx dy
E
Where E is an ellipse with center in the origin and semi-axes with lenght 1 and 2, that has the cartesian axes as axis of simmetry, 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 alon 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).

... a change of variables: ( 24) our integral becomes ( 25 ... This is an Euler ordinary differential equation (ODE) which ... ω v where V = a n-dimensional vector field. ...

... (1.28) Where C is some integration constant. ... q) moving with velocity v in the presence of magnetic field B and electric field E is ... For a general velocity vector. ...

... time, the actual displacement is the vector (or algebraic ... Then by integration, f1(ct) = f2(ct), which ... for simplicity that the electric field vectors of all ...

... Calculate the divergence and curl of the vector field F(x ... and z, we can calculate the gradient vectors to be ... integrate by parts to find the first two integrals: ...

... form ∇ × B = µ 0J , and the integral ∫ B dl ... Biot-Savart law, which typically involves vector quantities on ... the magnitude of the magnetic field for a ...

... values of the rr components of the vectors A and ... and electric fields or the magnetic vector potential and ... A.dl = ∫ At dl = 0 evaluating the integral we will ...

... In vector form, we write: ... Green's functions are widely used in electromagnetic and other field theories. ... up a number of point sources, ie by integration of the ...

A game reserve has two grass fields separated by a crocodile ... of a vector with respect to a vector is the ... If A is singular, the integral exists but must be ...

... had magnitude k/(x2 + y2) an inverse-square central force field. ... So equating the vector components on both sides of the equation... Finding an integration factor: ...

... red lines indicate examples of integral curves, the ... The vector field is the following: y. ... 1] VI Arnold, R. Cook, Ordinary differential equations, Springer, 1992. ...