Question

let
y1=e^x be a solution of the DE 2y''-5y'+3y=0 use the reduction of
order method to find a second linearly independent solution y2 of
the given DE

Answer #1

Given that y1=e3x is a solution of the DE y′′−4y′+3y=0 . Using
the reduction of order method a second solution of the equation is
y2=uy1 where u is ?

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 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...

($4.2 Reduction of Order):
(a) Let y1(x) = x be a solution of the homogeneous ODE xy′′
−(x+2)y′ + ((x+2)/x)y = 0. Use the reduction
of order to find a second solution y2(x), and write the general
solution.

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))/y12(x)dx (5)
as instructed, to find a second solution
y2(x).4x2y'' + y = 0; y1 = x1/2 ln(x)
y2 = ?

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'' + 64y = 0; y1 =
cos(8x)
y2 =

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 =

use reduction of order to find a second independent solution to
t2y''+2ty'-2y=0 with the first solution being given by
y1(t)=t.

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'' + 100y = 0;
y1 = cos 10x
I've gotten to the point all the way to where y2 = u y1, but my
integral is wrong for some reason
This was my answer
y2= c1((sin(20x)+20x)cos10x)/40 + c2(cos(10x))

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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