Question

Determine the type of below equations and solve it.

a-)(sin(xy)+xycos(xy)+2x)dx+(x^{2}cos(xy)+2y)dy=0

b-)(t-a)(t-b)y’-(y-c)=0 a,b,c are constant.

Answer #1

1. Solve the following differential equations.
(a) dy/dt +(1/t)y = cos(t) +(sin(t)/t) , y(2pie) = 1
(b)dy/dx = (2x + xy) / (y^2 + 1)
(c) dy/dx=(2xy^2 +1) / (2x^3y)
(d) dy/dx = y-x-1+(xiy+2) ^(-1)
2. A hollow sphere has a diameter of 8 ft. and is filled half way
with water. A circular hole (with a radius of 0.5 in.) is opened at
the bottom of the sphere. How long will it take for the sphere to
become empty?...

i)Please state if the following equations are exact or not:
(a) (sin(xy) − xy cos(xy))dx + x^2 cos(xy)dy = 0
(b) (x^3 + xy^2 )dx + (x^2 y + y^3 )dy = 0
ii) Determine if the following equation is exact, and if it is
exact, find its complete integral in the form g(x, y) = C:
(3(x)^2 + 2(y)^2 )dx + (4xy + 6(y)^2 )dy = 0

Solve the Initial Value Problem
(y2 cos(x) − 3x2y − 2x) dx + (2y sin(x) −
x3 + ln(y)) dy = 0, y(0) = e

Consider the system [ x' = -2y & y' = 2x] . Use dy/dx to
find the curves y = y(x).
Draw solution curves in the xy phase plane. What type of
equilibrium point is the origin?

Solve the equation.
(2x^3+xy)dx+(x^3y^3-x^2)dy=0
give answer in form F(x,y)=c

Solve:
(2x^2 - y) dx + (x + y^2) dy = 0

Use the Laplace transform to solve the given system of
differential equations. dx/dt=x-2y dy/dt=5x-y x(0) = -1, y(0) =
6

dx/dt=y, dy/dt=2x-2y

Initial value problem : Differential equations:
dx/dt = x + 2y
dy/dt = 2x + y
Initial conditions:
x(0) = 0
y(0) = 2
a) Find the solution to this initial value problem
(yes, I know, the text says that the solutions are
x(t)= e^3t - e^-t and y(x) = e^3t + e^-t
and but I want you to derive these solutions yourself using one
of the methods we studied in chapter 4) Work this part out on paper
to...

Obtain the general solution of
x2d2y/dx2 -2x dy/dx
+2y=sin(lnx)

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