Question

Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. x^2 − 3y^2 = 6, ϕ = 60°

Answer #1

A conic section is given by the equation 4x2 + 10xy +
4y2 = 36.
Use rotation of coordinate axes through an appropriate acute angle
θ to find the new equation of the conic section in the
uv-coordinate axes , where x = u cos(θ) - v sin(θ) , y = u sin(θ) +
v cos(θ). Then identify the conic section.

A conic section is given by the equation 57x^2+14√3
xy+43y^2-576=0, identity and sketch the conics.

Determine a region of the xy-plane for which the given
differential equation would have a unique solution whose graph
passes through a point (x_0, y_0) in the region. (1+y^3)y' =
x^2

determine if the xy-plane for which the given differential
equation would have a unique solution whose graph passes through
the point (x0,y0) in the region
dy/dx=y^(2/3)
x(dy/dx)=y

Determine an equation for the tangent to each function at the
point with the given x-coordinate.
a) f(x) = tanx , x = π/4 b) f(x) = 6tanx -
tan2x, x = 0
Doing limits using the formula
y = tanx --> dy/dx = sec^2x

Go through separation of variables solution of Laplace’s
equation in cylindrical coordinates when it is not legitimate to
ignore the z dependence. Determine what the functional
form is for the z and φ dependence. Check the solutions for the ρ
dependence.

Determine for which values of m the function ϕ(x)=e^mx is a
solution to the given equation. (a) (d^2 y)/(dx^2 )+6
dy/dx+5y=0

Find the equation(s) of the tangent line(s) when x = 0
for:
xy^5 +2(x^2)y−y^2 −3y−2=0

Q1). i). Given that ρs = x
3y + xy, calcultae ∮ ρs ds, over the region, y ≤ x
3
, 0 < ? < 1.
ii). Find the unit vector along the line joining point (-2, 8, 6)
to point ( -6 , -2, -4).
iii). Determine the value of charge moving with a certain
velocity
?̅= (5 ?̅̅?̅ + 6 ?̅̅?̅ − 3 ?̅̅?̅) × 103 m/s in a magnetic field
velocity?̅ = (3 ?̅̅?̅ − 4...

Without looking for the solutions in a power series around Xo
given, determine the radius of convergence for such solutions in
series for the differential equation:
(x^2 + x-12)y"+(x^2-1)y'+3y=0
and the values for Xo= -5, 0, 6

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