Question

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.

Answer #1

Given y1(t)=t^2 and y2(t)=t^-1 satisfy the corresponding
homogeneous equation of
t^2y''−2y=2−t3, t>0
Then the general solution to the non-homogeneous equation can be
written as y(t)=c1y1(t)+c2y2(t)+yp(t)
yp(t) =

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)=

Given that y1 = t, y2 = t 2 are solutions to the homogeneous
version of the nonhomogeneous DE below, verify that they form a
fundamental set of solutions. Then, use variation of parameters to
find the general solution y(t).
(t^2)y'' - 2ty' + 2y = 4t^2 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 = ?????

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))

The nonhomogeneous equation t2 y′′−2 y=29 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.

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

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.

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.

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

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