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y (^) 1 = cos 4 x y '' + 16 y = 0 y (^) 2
y 2 = sin ( x )
y '' 4 y ' + 3 y = 2 e
4 x
y 1 ( x ) = e
x
y 2 ( x )
y (^) 2 = e x
yp = e
x
y (^) 1 = e 2 x
y '' 4 y = 0 ( , )
y 2 y 1 , y 2 ( , )
y (^) 2 = e x
y 1 ( x )
y (^) 2 = y (^) 1 ( x ) e
P (^) ( x) dx
y
2
1 (^ x )
dx y 2 ( x )
9 y ' ' 12 y ' + 4 y = 0 ; y (^) 1 = e
2 x 3
y (^) 2 = e
^2 x 3 y (^) 2 = xe
x 3 y (^) 2 = e
2 x 3
y (^) 2 = xe
^ x 3 y (^) 2 = xe
2 x 3
1 Given that is a solution of , use reduction of order to find a second solution.
The second solution is.
2 Given the equation.
is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution of the homogeneous equation and a particular solution of the given nonhomogeneous equation.
The second solution of the associated homogeneous equation is. The particular solution of the
nonhomogeneous equation is.
3 Given that is a solution of on the interval , use reduction of order to find a second
solution so that is a linearly independent set on the interval.
The answer is.
4 The indicated function is a solution of the given differential equation. Use reduction of order or the formula
, as instructed, to find a second solution.
a. c. e.
b. d.
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Name: __________________ Class: Date: _____________
y 1 ( x )
y 2 = y 1 ( x ) e
P (^) ( x) dx
y 2
1 (^ x )
dx y 2 ( x )
x
2 y ' ' 3 xy ' + 5 y = 0 ; y (^) 1 = x
2
cos ln( x )
y (^) 2 = x
2
cos ln( x ) y^ 2 =^ x^ sin ln(^ x )^ y^ 2 =^ x^ cos ln(^ x )
y (^) 2 = x 3
cos ln( x ) y 2 = x
2
sin ln( x )
y 1 ( x )
y 2 = y 1 ( x ) e
P (^) ( x) dx
y
2
1 (^ x )
dx y 2 ( x )
1 x 2
( ) y ' ' + 2 xy ' = 0 ; y 1 = 1
y 1 ( x )
y 2 ( x )
y ' ' 4 y = 2 ; y (^) 1 = e
2 x
y 2 ( x )
y p ( x )
y 1 ( x )
y 2 ( x )
y ' ' 3 y ' + 2 y = 5 e 3 x ; y (^) 1 = e x
y 2 ( x )
y p ( x )
5 The indicated function is a solution of the given differential equation. Use reduction of order or the formula
to find a second solution.
a. c.^ e.
b. d.
6 The indicated function is a solution of the given differential equation. Use reduction of order or the formula
, as instructed, to find a second solution.
7 The indicated function is a solution of the associated homogeneous equation. Use the method of reduction of order to
find a second solution of the homogeneous equation and a particular solution of the given nonhomogeneous equation.
Find a second solution.
Find a particular solution.
8 The indicated function is a solution of the associated homogeneous equation. Use the method of reduction of order to
find a second solution of the homogeneous equation and a particular solution of the given nonhomogeneous equation.
Find a second solution.
Find a particular solution.
PAGE 2
Name: __________________ Class: Date: _____________
