Question

) Check that each of the following functions solves the corresponding differential equation, by computing both the left-hand side and right-hand side of the differential equation.

(a) y = cos2 (x) solves dy/dx = −2 sin(x) √y

(b) y = 4x + 1/x solves x dy dx + 2/x = y

(c) y = e x 2+3 solves dy/dx = 2xy

(d) y = ln(1 + x 2 ) solves e y dy dx = 2x

Answer #1

1- Find the solution of the following equations. For each
equation, 2- determine the type of the category that the equation
belongs to.
1. y/x cos y/x dx − ( x/y sin y/x + cos y/x ) dy = 0
2. x(1 − y^2 )dx + y(8 − x^2 )dy = 0
3. (x^2 − x + y^2 )dx − (e^y − 2xy)dy = 0
4. 2x sin 3ydx + 3x^2 cos 3ydy = 0
5. (x ln x −...

Check if the following functions satisfy the differential
equation dy/dx = x+y.
In each case, after your work is complete say whether the
function fits the equation.
a. y= -x-1
b. y= e^x -x
c. y=2e^x-x-1

Find the solution to the separable differential equation dy =
x cos2 y + sin x cos2 y satisfying π dx
the initial condition y = 4 when x = π.

The
following are solutions to homogeneous linear differential
equations. Obtain the corresponding differential equation with
real, constant cocfficients that is satisfied by the given
function:
a) y=4e^2x+3e^-x
b) y=x^2-5sin3x
c) y=-2x+1/2e^4x
d) y=xe^-xsin2x+3e^-xcos2x

solve the differential equation
(2y^2+2y+4x^2)dx+(2xy+x)dy=0

1) Solve the given differential equation by separation of
variables.
exy
dy/dx = e−y +
e−6x −
y
2) Solve the given differential
equation by separation of variables.
y ln(x) dx/dy = (y+1/x)^2
3) Find an explicit solution of the given initial-value
problem.
dx/dt = 7(x2 + 1), x( π/4)= 1

Which of the following functions are solutions of the
differential equation y″−3y′−10y=0 y ″ − 3 y ′ − 10 y = 0 ?
A. y(x)=exy(x)=ex
B. y(x)=e−xy(x)=e−x
C. y(x)=−2xy(x)=−2x
D. y(x)=e5xy(x)=e5x
E. y(x)=e−2xy(x)=e−2x
F. y(x)=0y(x)=0
G. y(x)=5xy(x)=5x

exact differential equation, (2xy+x)dx+(x^2+y)dy=0

Determine whether the equation is exact. If it is exact, find
the solution. If it is not, enter NS.
(y/x+7x)dx+(ln(x)−2)dy=0, x>0
Enclose arguments of functions in parentheses. For example,
sin(2x).

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?...

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