This problem demonstrates the variation-of-parameters method for first-order linear differential equations. Consider the first-order linear differential equation. y ' + p ( x ) y = q ( x ) ( 1.6.15 ) (a) Show that the general solution to the associated homogeneous equation y ' + p ( x ) y = 0 is y H ( x ) = c 1 e − ∫ p ( x ) d x (b) Determine the function u ( x ) such that y ( x ) = u ( x ) e − ∫ p ( x ) d x is a solution to (1.6.15) and hence derive the general solution to (1.6.15)
This problem demonstrates the variation-of-parameters method for first-order linear differential equations. Consider the first-order linear differential equation. y ' + p ( x ) y = q ( x ) ( 1.6.15 ) (a) Show that the general solution to the associated homogeneous equation y ' + p ( x ) y = 0 is y H ( x ) = c 1 e − ∫ p ( x ) d x (b) Determine the function u ( x ) such that y ( x ) = u ( x ) e − ∫ p ( x ) d x is a solution to (1.6.15) and hence derive the general solution to (1.6.15)
Solution Summary: The author explains that the general solution to the homogeneous equation is y'+p(x)y = q left.
This problem demonstrates the variation-of-parameters method for first-order linear differential equations. Consider the first-order linear differential equation.
y
'
+
p
(
x
)
y
=
q
(
x
)
(
1.6.15
)
(a) Show that the general solution to the associated homogeneous equation
y
'
+
p
(
x
)
y
=
0
is
y
H
(
x
)
=
c
1
e
−
∫
p
(
x
)
d
x
(b) Determine the function
u
(
x
)
such that
y
(
x
)
=
u
(
x
)
e
−
∫
p
(
x
)
d
x
is a solution to (1.6.15) and hence derive the general solution to (1.6.15)
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