Question

Obtain the general solution of the following equations:

x^2Uxx +2xyUxy + y^2Uyy + xyUx + y^2Uy = 0,

Answer #1

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

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

3. Find the general solution to each of the following
differential equations.
(a) y'' - 3y' + 2y = 0
(b) y'' - 10y' = 0
(c) y'' + y' - y = 0
(d) y'' + 2y' + y = 0

For each of the following equations, find the general
solution
y''+y=tcost

solve the following system of differential equations
and find the general solution
(D+3)x+(D-1)y=0 and 2x+(D-3)y=0
please show the steps

Find the general solution of the following nonhomogeneous
equations:
• y '' − 2y ' + 5y = 25x ^2 + 12
• y ''− 3y ' + 2y = 10 sen 2x
• y '' − 2y ' + y = e^x + e^−2x
• y '' − y = x sen x
• y '' − 6y ' + 9y = 5e^x sen x
• y '' + y ' + y + 1 = sen 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...

Solve the following system of equations and select the correct
general solution from below.
x’ = -2y and y’ = (1/2) x

Find the general solution of the following differential
equations. Primes denote derivates with respect to x.
1) x(x+3y)y'= y(x-3y)
2) 3xy^2y'= 21x^3+3y^3
3) x^2y'= xy+10y^2
4) x(4x+3y)y'+ y(12x+3y)= 0
5)2xyy' = 2y^2 + 7xsqrt(9x^2+y^2)

(61). (Bernoulli’s Equation): Find the general solution of the
following first-order differential equations:(a) x(dy/dx)+y=
y^2+ln(x) (b) (1/y^2)(dy/dx)+(1/xy)=1

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