Question

Follow the steps below to use the method of reduction of order
to find a second solution y_{2} given the following
differential equation and y_{1}, which solves the given
homogeneous equation:

xy" + y' = 0; y_{1} = ln(x)

Step #1: Let y_{2} = uy_{1}, 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 first order differential equation in w.

Step #4: Solve for w and thus u' in your previous answer.

Step #5: Solve for u from your previous equation for u'.

Step #6: Solve for y_{2} using our definition in the
first step, (use c_{1} = -1 and c_{2} = 0, which
are assumed to be the constants of integration).

Step #7: Verify that y_{2} is a solution to the original
given differential equation, and check that W[y_{1},
y_{2}] does not equal zero in *I* := R.

Answer #1

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'.

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
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 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 = ?

Use the method of variation of parameters to determine the
general solution of the given differential equation.
y′′′−y′=3t
Use C1, C2, C3, ... for the constants of integration.

Solve the following differential equation by variation of
parameters. Fully evaluate all integrals.
y′′+9y=sec(3x).
a. Find the most general solution to the associated homogeneous
differential equation. Use c1 and c2 in your
answer to denote arbitrary constants, and enter them as c1 and
c2.
b. Find a particular solution to the nonhomogeneous differential
equation y′′+9y=sec(3x).
c. Find the most general solution to the original nonhomogeneous
differential equation. Use c1 and c2 in your
answer to denote arbitrary constants.

Use variation of parameters to find a general solution to the
differential equation given that the functions y1 and y2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
y1=et y2=t+1
ty''-(t+1)y'+y=2t2

1) find a solution for a given differential equation
y1'=3y1-4y2+20cost ->y1 is not y*1 & y2 is not y*2
y2'=y1-2y2
y1(0)=0,y2(0)=8
2)by setting y1=(theta) and y2=y1', convert the following 2nd
order differential equation into a first order system of
differential equations(y'=Ay+g)
(theta)''+4(theta)'+10(theta)=0

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)

Use variation of parameters to find a general solution to the
differential equation given that the functions y 1 and y 2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
ty"-(t+1)y'+y=30t^2 ; y1=e^t , y2=t+1
The general solution is y(t)= ?

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