Question

Determine the region in which the given equation is hyperbolic,
parabolic,

or elliptic, and transform the equation in the respective region to
canonical

form.

x^2Uxx − 2xyUxy + y^2Uyy = e^x

Answer #1

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

Match each equation with its name. (calculus 3)
(x/7)^2-(y/9)^2=z/4
(x/7)^2-y/9+(z/4)^2=0
−x/7+(y/9)^2+(z/4)^2=0
(x/7)^2+(y/9)^2=(z/4)^2
(x/7)^2+(y/7)^2+(z/7)^2=1
Hyperbolic paraboloid
Elliptic paraboloid on x axis
Cone on z axis
Elliptic paraboloid on y axis
Sphere

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

Use the substitution x = et to transform the given Cauchy-Euler
equation to a differential equation with constant coefficients.
(Use yp for dy /dt and ypp for d2y/dt2 .) x2y'' + 10xy' + 8y =
x2
Solve the original equation by solving the new equation using
the procedures in Sections 4.3-4.5. y(x) =

Consider the second order linear partial differential equation
a(x, y)uxx + 2b(x, y)uxy + c(x, y)uyy + d(x, y)ux + e(x, y)uy +
f(x, y)u = 0. (1)
a). Show that under a coordinate transformation ξ = ξ(x, y), η =
η(x, y) the PDE (1) transforms into the PDE α(ξ, η)uξξ + 2β(ξ,
η)uξη + γ(ξ, η)uηη + δ(ξ, η)uξ + (ξ, η)uη + ψ(ξ, η)u = 0, (2) where
α(ξ, η) = aξ2 x + 2bξxξy + cξ2...

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

1) Solve the given differential equation by using an appropriate
substitution. The DE is a Bernoulli equation.
x dy/dx +y= 1/y^2
2)Consider the following differential equation.
(25 − y2)y' = x2
Let f(x, y) = x^2/ 25-y^2. Find the derivative of f.
af//ay=
Determine a region of the xy-plane for which the given
differential equation would have a unique solution whose graph
passes through a point
(x0, y0) in the region.
a) A unique solution exists in the region consisting...

Determine the equation of the line in general form given the
following information.
The equation of a line perpendicular to 3x-5y+19=0 and has an
x-intercept of 2

consider the region E, which is under the surface z=8-(x^2+y^2) and
above the region R in the xy-plane bounded by x^2+y^2=4.
a) sketch the solid region E and the shadow it casts in the
xy-plane
b) find the mass of E if the density is given by
δ(x,y,z)=z

#3
Determine the form of a particular solution of equations (do not
solve equation):
a) y’’ – 2y’ + y = e^x
b) y’’ + y = 4x + 10sinx

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