Determining “trial” solution for particular integral
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In the DE $x^2y''+xy'-4y=15x^3$, supposedly the particular integral can be found by attempting $y=Cx^3$ and then substituting. However it's not clear to me why this is. I would have thought the most logical starting point would be an arbitrary polynomial of degree $3$, i.e. $Ax^3+Bx^2+Cx+D$. How does one know that the true solution will only be of the form $Cx^3$?
ordinary-differential-equations
asked Jan 8 at 21:53
cb7cb7
1476
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add a comment |
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In the DE $x^2y''+xy'-4y=15x^3$, supposedly the particular integral can be found by attempting $y=Cx^3$ and then substituting. However it's not clear to me why this is. I would have thought the most logical starting point would be an arbitrary polynomial of degree $3$, i.e. $Ax^3+Bx^2+Cx+D$. How does one know that the true solution will only be of the form $Cx^3$?
ordinary-differential-equations
asked Jan 8 at 21:53
cb7cb7
1476
$endgroup$
$begingroup$
Will $A$ have a chance at influencing $B$? Are any of $A,B,C,D$ coupled after inserting the full polynomial?
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– LutzL
Jan 8 at 21:59
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May I confess that I would have done the same as you ?
$endgroup$
– Claude Leibovici
Jan 9 at 6:20
add a comment |
$begingroup$
In the DE $x^2y''+xy'-4y=15x^3$, supposedly the particular integral can be found by attempting $y=Cx^3$ and then substituting. However it's not clear to me why this is. I would have thought the most logical starting point would be an arbitrary polynomial of degree $3$, i.e. $Ax^3+Bx^2+Cx+D$. How does one know that the true solution will only be of the form $Cx^3$?
ordinary-differential-equations
asked Jan 8 at 21:53
cb7cb7
1476
$endgroup$
In the DE $x^2y''+xy'-4y=15x^3$, supposedly the particular integral can be found by attempting $y=Cx^3$ and then substituting. However it's not clear to me why this is. I would have thought the most logical starting point would be an arbitrary polynomial of degree $3$, i.e. $Ax^3+Bx^2+Cx+D$. How does one know that the true solution will only be of the form $Cx^3$?
ordinary-differential-equations
ordinary-differential-equations
asked Jan 8 at 21:53
cb7cb7
1476
asked Jan 8 at 21:53
cb7cb7
1476
asked Jan 8 at 21:53
cb7cb7
1476
asked Jan 8 at 21:53
cb7cb7
1476
asked Jan 8 at 21:53
cb7cb7
1476
1476
$begingroup$
Will $A$ have a chance at influencing $B$? Are any of $A,B,C,D$ coupled after inserting the full polynomial?
$endgroup$
– LutzL
Jan 8 at 21:59
$begingroup$
May I confess that I would have done the same as you ?
$endgroup$
– Claude Leibovici
Jan 9 at 6:20
add a comment |
$begingroup$
Will $A$ have a chance at influencing $B$? Are any of $A,B,C,D$ coupled after inserting the full polynomial?
$endgroup$
– LutzL
Jan 8 at 21:59
$begingroup$
May I confess that I would have done the same as you ?
$endgroup$
– Claude Leibovici
Jan 9 at 6:20
$begingroup$
Will $A$ have a chance at influencing $B$? Are any of $A,B,C,D$ coupled after inserting the full polynomial?
$endgroup$
– LutzL
Jan 8 at 21:59
$begingroup$
Will $A$ have a chance at influencing $B$? Are any of $A,B,C,D$ coupled after inserting the full polynomial?
$endgroup$
– LutzL
Jan 8 at 21:59
$begingroup$
May I confess that I would have done the same as you ?
$endgroup$
– Claude Leibovici
Jan 9 at 6:20
$begingroup$
May I confess that I would have done the same as you ?
$endgroup$
– Claude Leibovici
Jan 9 at 6:20
add a comment |
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$begingroup$
Will $A$ have a chance at influencing $B$? Are any of $A,B,C,D$ coupled after inserting the full polynomial?
$endgroup$
– LutzL
Jan 8 at 21:59
$begingroup$
May I confess that I would have done the same as you ?
$endgroup$
– Claude Leibovici
Jan 9 at 6:20