Question

Solve using reduction of order or variation of parameters

4x^{2}y” + 4xy’ + (4x^{2} – 1)y = F(x)

y_{1}(x) = x ^{-1/2} sin
x and
y_{2}(x) = x ^{-1/2} cos
x

Your answer will come out in terms of integrals involving F(x).

Find a function F(x) for which you can easily calculate the necessary integrals in the answer and do the integral

Answer #1

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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 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 = ?

Solve y’’ – 11y’ + 24y = ex +3x using:
Reduction of order
V.C superposition
Variation of parameters

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...

solve the given differential equation by variation of
parameters.
y”+y=1/cos(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'' + 64y = 0; y1 =
cos(8x)
y2 =

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

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

Solve the differential equation by variation of parameters.
y'' + 3y' + 2y = 1 / (7 + e^x)

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