Question

Verify that the functions y1 = cos x − cos 2x and y2 = sin x − cos 2x both satisfy the differential equation y′′ + y = 3 cos 2x.

Answer #1

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In this problem verify that the given functions y1 and y2
satisfy the corresponding homogeneous equation. Then find a
particular solution of the nonhomogeneous equation.
x^2y′′−3xy′+4y=31x^2lnx, x>0, y1(x)=x^2, y2(x)=x^2lnx. Enter an
exact answer.

y1 = 2 cos(x) − 1 is a particular solution for y'' + 4y = 6
cos(x) − 4. y2 = sin(x) is a particular solution for y''+4y = 3
sin(x). Using the two particular solutions, find a particular
solution for y''+4y = 2 cos(x)+sin(x)− 4/3 . Verify if the
particular solution satisfies the given DE.
[Hint: Rewrite the right hand of this equation in terms of the
given particular solutions to get the particular solution] Verify
if the particular...

Show that the given functions y1 and y2 are solutions to the DE.
Then show that y1 and y2 are linearly independent. write the
general solution. Impose the given ICs to find the particular
solution to the IVP.
y'' + 25y = 0; y1 = cos 5x; y2 = sin 5x; y(0) = -2; y'(0) =
3.

Verify that the given functions form a fundamental set of
solutions of the differential equation on the indicated interval.
Form the general solution.
1.) y'' − 4y = 0; cosh 2x, sinh 2x, (−∞,∞)
2.) y^(4) + y'' = 0; 1, x, cos x, sin x (−∞,∞)

Consider the equation y'' + 4y = 0.
a) Justify why the functions y1 = cos(4t) and y2 = sin(4t) do not
constitute a fundamental set of solutions of the above
equation.
b) Find y1, y2 that constitute a fundamental set of solutions,
justifying your answer.

The wave functions
y1(x, t) = (0.150 m)sin(3.00x − 1.50t)
and
y2(x, t) = (0.250 m)cos(6.00x − 3.00t)
describe two waves superimposed on a string, with x and
y in meters and t in seconds. What is the
displacement y of the resultant wave at the following.
(Include the sign of the value in your answers.)
(a)
x = 0.700 m
and
t = 0
m
(b)
x = 1.15 m
and
t = 1.15 s
m
(c)
x =...

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

Two waves on one string are described by the wave functions
y1
=
2.5 cos(3.5x − 1.3t)
y2
=
3.5 sin(4.5x − 2.5t),
where x and y are in centimeters and
t is in seconds. Find the values of the waves
y1 +
y2
at the following points. (Remember that the arguments of the
trigonometric functions are in radians.)
(a)x = 1.00, t = 1.00
(b) x = 1.00, t = 0.500
(c) x = 0.500, t = 0

Consider the differential equation:
66t^2y''+12t(t-11)y'-12(t-11)y=5t^3, . You can verify that y1 = 5t
and y2 = 4te^(-2t/11)satisfy the corresponding homogeneous
equation.
The Wronskian W between y1 and y2 is W(t) =
(-40/11)t^2e^((-2t)/11)
Apply variation of parameters to find a particular solution.
yp = ?????

Two waves on one string are described by the wave functions
y1 = 2.5 cos(4.5x −
1.3t)
y2 = 4.5 sin(5.5x − 1.5t)
where x and y are in centimeters and
t is in seconds. Find the superposition of the waves
y1 + y2 at the following
points. (Remember that the arguments of the trigonometric functions
are in radians.)
(a) x = 1.00, t = 1.00
cm
(b) x = 1.00, t = 0.500
cm
(c) x = 0.500, t...

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