Is 2nd-order ODE with quadratic coefficients solvable?
$begingroup$
Consider an ODE eigensystem
$$begin{bmatrix}
0 & d_1-mathrm id_2 \
d_1+mathrm id_2 & 0
end{bmatrix}
begin{bmatrix} a(y) \ b(y) end{bmatrix} = lambda begin{bmatrix} a(y) \ b(y) end{bmatrix},
$$
where $$d_1=-mathrm i(p+qy)partial_y+ry+s$$ $$d_2=-mathrm i(u+vy)partial_y+wy+t,$$ $p,q,r,s,u,v,w,t$ are just real constants, and $mathrm i$ is the imaginary unit. Is it analytically solvable?
I reduce it to a 2nd-order ODE of $b$ with coefficients quadratic in $y$
$$alpha b''(y) + beta b'(y) + gamma b(y)=-lambda^2 b(y)$$
where $$alpha=(p+q y)^2+(u+v y)^2$$
$$beta=p (q+2 i s-i v)+u (v+iq+2 it)+(2 i p r+q^2+2 i q s+2 i t v+2 i u w+v^2)y+2 i (q r+v w)y^2 $$
$$gamma=-s^2-t^2+(p+i u) (w+i r)+[w (q-2 t+i v)-r (-i q+2 s+v)]y-(r^2+w^2)y^2 $$
When $u,v=0$ or $p,q=0$, it is solvable, although the coefficients are still quadratic polynomials of $y$. I was wondering if the more general case could be tackled as well. But I don't know how to proceed.
ordinary-differential-equations eigenvalues-eigenvectors sturm-liouville
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
$endgroup$
add a comment |
$begingroup$
Consider an ODE eigensystem
$$begin{bmatrix}
0 & d_1-mathrm id_2 \
d_1+mathrm id_2 & 0
end{bmatrix}
begin{bmatrix} a(y) \ b(y) end{bmatrix} = lambda begin{bmatrix} a(y) \ b(y) end{bmatrix},
$$
where $$d_1=-mathrm i(p+qy)partial_y+ry+s$$ $$d_2=-mathrm i(u+vy)partial_y+wy+t,$$ $p,q,r,s,u,v,w,t$ are just real constants, and $mathrm i$ is the imaginary unit. Is it analytically solvable?
I reduce it to a 2nd-order ODE of $b$ with coefficients quadratic in $y$
$$alpha b''(y) + beta b'(y) + gamma b(y)=-lambda^2 b(y)$$
where $$alpha=(p+q y)^2+(u+v y)^2$$
$$beta=p (q+2 i s-i v)+u (v+iq+2 it)+(2 i p r+q^2+2 i q s+2 i t v+2 i u w+v^2)y+2 i (q r+v w)y^2 $$
$$gamma=-s^2-t^2+(p+i u) (w+i r)+[w (q-2 t+i v)-r (-i q+2 s+v)]y-(r^2+w^2)y^2 $$
When $u,v=0$ or $p,q=0$, it is solvable, although the coefficients are still quadratic polynomials of $y$. I was wondering if the more general case could be tackled as well. But I don't know how to proceed.
ordinary-differential-equations eigenvalues-eigenvectors sturm-liouville
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
$endgroup$
add a comment |
$begingroup$
Consider an ODE eigensystem
$$begin{bmatrix}
0 & d_1-mathrm id_2 \
d_1+mathrm id_2 & 0
end{bmatrix}
begin{bmatrix} a(y) \ b(y) end{bmatrix} = lambda begin{bmatrix} a(y) \ b(y) end{bmatrix},
$$
where $$d_1=-mathrm i(p+qy)partial_y+ry+s$$ $$d_2=-mathrm i(u+vy)partial_y+wy+t,$$ $p,q,r,s,u,v,w,t$ are just real constants, and $mathrm i$ is the imaginary unit. Is it analytically solvable?
I reduce it to a 2nd-order ODE of $b$ with coefficients quadratic in $y$
$$alpha b''(y) + beta b'(y) + gamma b(y)=-lambda^2 b(y)$$
where $$alpha=(p+q y)^2+(u+v y)^2$$
$$beta=p (q+2 i s-i v)+u (v+iq+2 it)+(2 i p r+q^2+2 i q s+2 i t v+2 i u w+v^2)y+2 i (q r+v w)y^2 $$
$$gamma=-s^2-t^2+(p+i u) (w+i r)+[w (q-2 t+i v)-r (-i q+2 s+v)]y-(r^2+w^2)y^2 $$
When $u,v=0$ or $p,q=0$, it is solvable, although the coefficients are still quadratic polynomials of $y$. I was wondering if the more general case could be tackled as well. But I don't know how to proceed.
ordinary-differential-equations eigenvalues-eigenvectors sturm-liouville
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
$endgroup$
Consider an ODE eigensystem
$$begin{bmatrix}
0 & d_1-mathrm id_2 \
d_1+mathrm id_2 & 0
end{bmatrix}
begin{bmatrix} a(y) \ b(y) end{bmatrix} = lambda begin{bmatrix} a(y) \ b(y) end{bmatrix},
$$
where $$d_1=-mathrm i(p+qy)partial_y+ry+s$$ $$d_2=-mathrm i(u+vy)partial_y+wy+t,$$ $p,q,r,s,u,v,w,t$ are just real constants, and $mathrm i$ is the imaginary unit. Is it analytically solvable?
I reduce it to a 2nd-order ODE of $b$ with coefficients quadratic in $y$
$$alpha b''(y) + beta b'(y) + gamma b(y)=-lambda^2 b(y)$$
where $$alpha=(p+q y)^2+(u+v y)^2$$
$$beta=p (q+2 i s-i v)+u (v+iq+2 it)+(2 i p r+q^2+2 i q s+2 i t v+2 i u w+v^2)y+2 i (q r+v w)y^2 $$
$$gamma=-s^2-t^2+(p+i u) (w+i r)+[w (q-2 t+i v)-r (-i q+2 s+v)]y-(r^2+w^2)y^2 $$
When $u,v=0$ or $p,q=0$, it is solvable, although the coefficients are still quadratic polynomials of $y$. I was wondering if the more general case could be tackled as well. But I don't know how to proceed.
ordinary-differential-equations eigenvalues-eigenvectors sturm-liouville
ordinary-differential-equations eigenvalues-eigenvectors sturm-liouville
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
edited Jan 21 at 18:26
xiaohuamao
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
asked Jan 21 at 7:24
xiaohuamaoxiaohuamao
304111
304111
add a comment |
add a comment |
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