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rookie404
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1.Homework Statement
We know the derivation of the method of variation of parameters for second order scalar differential. The task is to derive the method of variation of parameters for scalar equations using this approach: first convert the scalar equation into the first order system and then apply the method of variation of parameters for systems.
Hint: Assume that the fundamental set of solutions {y1(t), y2(t)} for your scalar equation is known. How would you construct a fundamental matrix for your system from the scalar fundamental set?
(my work, not in problem statement)
y''+p(t)y'+q(t)y=g(t) <--Second order nonhomogenous scalar
y' = P(t)y + g(t) <--first order nonhomogenous system
y''+p(t)y'+q(t)y=g(t)
change of variables...
z(t)= [ z1(t)
z2(t) ]
z1(t) = y(t)
z2(t) = y'(t)
z'2(t)+p(t)z2(t)+q(t)z1(t)=g(t)
z'1(t) = y'(t) = z2(t)
z'2(t) = y''(t) = -z2(t)p(t)-z1(t)q(t)+g(t)
P(t) =
[ 0 1
-q(t) -p(t) ]
G(t) =
[ 0
g(t) ]
y' = P(t)y + g(t) <--now what I have has this form (first order system)
Now I have the variation of parameters for systems derivation in front of me, the end result of which is:
y(t) = [tex]\psi[/tex](t)u0 + [tex]\psi[/tex](t)[tex]\int_t_0^t[/tex][tex]\psi[/tex]-1(s)g(s)ds
So I guess the hard part now is finding what the fundamental matrix [tex]\psi[/tex] is. I know the fundamental set is {y1, y2}
In the variation of parameters for second order differential equations we make the assumption y1u'1 + y2u'2 = 0. I am not supposed to make this assumption of this problem.
We know the derivation of the method of variation of parameters for second order scalar differential. The task is to derive the method of variation of parameters for scalar equations using this approach: first convert the scalar equation into the first order system and then apply the method of variation of parameters for systems.
Hint: Assume that the fundamental set of solutions {y1(t), y2(t)} for your scalar equation is known. How would you construct a fundamental matrix for your system from the scalar fundamental set?
Homework Equations
(my work, not in problem statement)
y''+p(t)y'+q(t)y=g(t) <--Second order nonhomogenous scalar
y' = P(t)y + g(t) <--first order nonhomogenous system
The Attempt at a Solution
y''+p(t)y'+q(t)y=g(t)
change of variables...
z(t)= [ z1(t)
z2(t) ]
z1(t) = y(t)
z2(t) = y'(t)
z'2(t)+p(t)z2(t)+q(t)z1(t)=g(t)
z'1(t) = y'(t) = z2(t)
z'2(t) = y''(t) = -z2(t)p(t)-z1(t)q(t)+g(t)
P(t) =
[ 0 1
-q(t) -p(t) ]
G(t) =
[ 0
g(t) ]
y' = P(t)y + g(t) <--now what I have has this form (first order system)
Now I have the variation of parameters for systems derivation in front of me, the end result of which is:
y(t) = [tex]\psi[/tex](t)u0 + [tex]\psi[/tex](t)[tex]\int_t_0^t[/tex][tex]\psi[/tex]-1(s)g(s)ds
So I guess the hard part now is finding what the fundamental matrix [tex]\psi[/tex] is. I know the fundamental set is {y1, y2}
In the variation of parameters for second order differential equations we make the assumption y1u'1 + y2u'2 = 0. I am not supposed to make this assumption of this problem.