Question

Use the METHOD of REDUCTION OF ORDER to find the general solution of the differential equation y"-4y=2 given that y1=e^-2x is a solution for the associated differential equation. When solving, use y=y1u and w=u'.

Answer #1

Use the method of reduction of order to find the
general solution of the following differential equation. (t^2)
d^2y/dt^2 + t dy/dt + (t^2-1/4) y = 0, y1(t) = sin t/sqrt(t)

Follow the steps below to use the method of reduction of order
to find a second solution y2 given the following
differential equation and y1, which solves the given
homogeneous equation:
xy" + y' = 0; y1 = ln(x)
Step #1: Let y2 = uy1, for u = u(x), and
find y'2 and y"2.
Step #2: Plug y'2 and y"2 into the
differential equation and simplify.
Step #3: Use w = u' to transform your previous answer into a
linear...

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

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
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

Use the provided solution and reduction of order to find the
general solution to the following differential equation.
x^(2)y''-6xy'+12y=0 y1(x)=x^4

Use the method of undetirmined coefficients to find the general
solution of the differential equation
y''+4y'-5y = 5cos(2x)

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))/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).
x2y'' -11xy' + 36y = 0; y1 = x6
y2 =

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