Question

Consider the differential equation x^{2}y''+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 y_{2}(x) =
v(x)y_{1}(x)]

Answer #1

Verify that the given function is a solution and use Reduction
of Order to find a second linearly independent solution.
a. x2y′′ −2xy′ −4y = 0, y1(x) = x4.
b. xy′′ − y′ + 4x3y = 0, y1(x) =
sin(x2).

Consider the differential equation
4x2y′′ − 8x2y′ + (4x2 + 1)y = 0
(a) Verify that x0 = 0 is a regular singular point of the
differential equation and then find one solution as a Frobenius
series centered at x0 = 0. The indicial equation has a single root
with multiplicity two. Therefore the differential equation has only
one Frobenius series solution. Write your solution in terms of
familiar elementary functions.
(b) Use Reduction of Order to find a second...

The indicated function y1(x) is a solution of the given
differential equation. Use reduction of order or formula (5) in
Section 4.2, y2 = y1(x) e−∫P(x) dx y 2 1 (x)
dx (5) as instructed, to find a second solution y2(x).
x2y'' -11xy' + 36y = 0; y1 = x6
y2 =

The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order, to find a
second solution dx **Please do not solve this via the
formula--please use the REDUCTION METHOD ONLY.
y2(x)= ??
Given: y'' + 2y' + y = 0; y1 =
xe−x

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))

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

Solve the 2nd Order
Differential Equation using METHOD OF REDUCTION
Please don't skip
steps!
(x-1)y"-xy'+y=0 x>1
y1(x)=x
My professor is
getting y2(x)=e^x and I don't understand how!

The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order or formula (5)
in Section 4.2,
y2 = y1(x)
e−∫P(x) dx
y
2
1
(x)
dx (5) as instructed, to find a
second solution y2(x).
y'' + 36y = 0; y1 =
cos(6x)
y2 =
2) The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order or formula (5)
in Section 4.2,
y2 = y1(x)
e−∫P(x) dx
y
2
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

Solve the 2nd Order Differential Equation using METHOD OF
REDUCTION
Please don't skip steps!
(x-1)y"-xy'+y=0 x>1 y1(x)=x

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