Particular integral for inhomogeneous term $cos(ax)sin(bx)$?
$begingroup$
If we have a second order ODE with the term in the question as its inhomogeneous component, what trial solutions are suggested for finding the particular integral?
I have tried to 'complexify' the ODE by trying the trial solution $cos(ax)e^{ibx}$ (with the intention of taking the imaginary part of the resulting particular integral), but so far it's taken 3 pages of algebra and I have yet to reach an answer. I struggle to believe there isn't a more elegant solution.
NB $a$ and $b$ are constants.
ordinary-differential-equations
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
$endgroup$
add a comment |
$begingroup$
If we have a second order ODE with the term in the question as its inhomogeneous component, what trial solutions are suggested for finding the particular integral?
I have tried to 'complexify' the ODE by trying the trial solution $cos(ax)e^{ibx}$ (with the intention of taking the imaginary part of the resulting particular integral), but so far it's taken 3 pages of algebra and I have yet to reach an answer. I struggle to believe there isn't a more elegant solution.
NB $a$ and $b$ are constants.
ordinary-differential-equations
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
$endgroup$
add a comment |
$begingroup$
If we have a second order ODE with the term in the question as its inhomogeneous component, what trial solutions are suggested for finding the particular integral?
I have tried to 'complexify' the ODE by trying the trial solution $cos(ax)e^{ibx}$ (with the intention of taking the imaginary part of the resulting particular integral), but so far it's taken 3 pages of algebra and I have yet to reach an answer. I struggle to believe there isn't a more elegant solution.
NB $a$ and $b$ are constants.
ordinary-differential-equations
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
$endgroup$
If we have a second order ODE with the term in the question as its inhomogeneous component, what trial solutions are suggested for finding the particular integral?
I have tried to 'complexify' the ODE by trying the trial solution $cos(ax)e^{ibx}$ (with the intention of taking the imaginary part of the resulting particular integral), but so far it's taken 3 pages of algebra and I have yet to reach an answer. I struggle to believe there isn't a more elegant solution.
NB $a$ and $b$ are constants.
ordinary-differential-equations
ordinary-differential-equations
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
edited Jan 5 at 0:14
Kenny Wong
19.3k21441
19.3k21441
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
asked Jan 5 at 0:10
Pancake_SenpaiPancake_Senpai
26017
26017
add a comment |
add a comment |
1 Answer
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$begingroup$
You can use the trigonometric identity,
$$ cos(ax)sin(bx) = frac 1 2 sin ((a + b)x) - frac 1 2 sin((a - b)x).$$
The right-hand side is free of annoying products!
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
You can use the trigonometric identity,
$$ cos(ax)sin(bx) = frac 1 2 sin ((a + b)x) - frac 1 2 sin((a - b)x).$$
The right-hand side is free of annoying products!
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
$endgroup$
add a comment |
$begingroup$
You can use the trigonometric identity,
$$ cos(ax)sin(bx) = frac 1 2 sin ((a + b)x) - frac 1 2 sin((a - b)x).$$
The right-hand side is free of annoying products!
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
$endgroup$
add a comment |
$begingroup$
You can use the trigonometric identity,
$$ cos(ax)sin(bx) = frac 1 2 sin ((a + b)x) - frac 1 2 sin((a - b)x).$$
The right-hand side is free of annoying products!
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
$endgroup$
You can use the trigonometric identity,
$$ cos(ax)sin(bx) = frac 1 2 sin ((a + b)x) - frac 1 2 sin((a - b)x).$$
The right-hand side is free of annoying products!
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
answered Jan 5 at 0:13
Kenny WongKenny Wong
19.3k21441
19.3k21441
add a comment |
add a comment |
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