Question

The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, ∞). Find the general solution of the given nonhomogeneous equation.

x2y'' + xy' + y = sec(ln(x))

y1 = cos(ln(x)), y2 = sin(ln(x))

Answer #1

The indicated function y1(x) is a solution of the associated
homogeneous differential equation. Use the method of reduction of
order to find a second solution y2(x) and a particular solution of
the given nonhomoegeneous equation.
y'' − y' = e^x
y1 = e^x

Use variation of parameters to find a general solution to the
differential equation given that the functions y1 and y2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
y1=et y2=t+1
ty''-(t+1)y'+y=2t2

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.

Use variation of parameters to find a general solution to the
differential equation given that the functions y 1 and y 2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
ty"-(t+1)y'+y=30t^2 ; y1=e^t , y2=t+1
The general solution is y(t)= ?

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 differential equation x2y''+xy'-y=0,
x>0.
a. Verify that y(x)=x is a solution.
b. Find a second linearly independent solution using the method
of reduction of order. [Please use y2(x) =
v(x)y1(x)]

The nonhomogeneous equation t2 y′′−2 y=19
t2−1, t>0, has homogeneous solutions
y1(t)=t2, y2(t)=t−1. Find the particular
solution to the nonhomogeneous equation that does not involve any
terms from the homogeneous solution.
Enter an exact answer.
Enclose arguments of functions in parentheses. For example,
sin(2x).
y(t)=

B. a non-homogeneous differential equation, a complementary
solution, and a particular solution are given. Find a solution
satisfying the given initial conditions.
y''-2y'-3y=6 y(0)=3 y'(0) = 11 yc=
C1e-x+C2e3x
yp = -2
C. a third-order homogeneous linear equation and three linearly
independent solutions are given. Find a particular solution
satisfying the given initial conditions
y'''+2y''-y'-2y=0, y(0) =1, y'(0) = 2, y''(0) = 0
y1=ex, y2=e-x,,
y3= e-2x

Consider the second-order homogeneous linear equation
y''−6y'+9y=0.
(a) Use the substitution y=e^(rt) to attempt to find two
linearly independent solutions to the given equation.
(b) Explain why your work in (a) only results in one linearly
independent solution, y1(t).
(c) Verify by direct substitution that y2=te^(3t) is a solution
to y''−6y'+9y=0. Explain why this function is linearly independent
from y1 found in (a).
(d) State the general solution to the given equation

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.

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