The kovacicsols command finds Louivillian solutions of ordinary linear homogeneous second-order differential equations of the form
a y″+b y′+c y=0, (12) |
where a, b and c are rational functions of the independent variable. kovacicsols uses Kovacic’s algorithm.
y=C1 y1+C2 y2 |
y2=y1 | ∫ | y1−2. (13) |
Examples.
y″= | ⎛ ⎜ ⎜ ⎝ |
| − |
| ⎞ ⎟ ⎟ ⎠ | y. |
⎡ ⎢ ⎣ | x |
| e |
| ,x |
| e |
| ⎤ ⎥ ⎦ |
x″(t)+ |
| x(t)=0. |
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ | ⎛ ⎜ ⎝ | −t | ⎛ ⎝ | t−1 | ⎞ ⎠ | ⎛ ⎜ ⎝ | 2 | √ |
| +2 t−1 | ⎞ ⎟ ⎠ | ⎞ ⎟ ⎠ |
| , | ⎛ ⎜ ⎝ | t | ⎛ ⎝ | t−1 | ⎞ ⎠ | ⎛ ⎜ ⎝ | 2 | √ |
| −2 t+1 | ⎞ ⎟ ⎠ | ⎞ ⎟ ⎠ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ |
|
y″= |
| y. |
r:=(4x^6-8x^5+12x^4+4x^3+7x^2-20x+4)/(4x^4) |
kovacicsols(y’’=r*y) |
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ |
y″+y′= |
| . |
⎡ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎦ |
y″+(19−x2) y=0 |
⎡ ⎢ ⎢ ⎣ | ⎛ ⎜ ⎜ ⎝ |
| x− |
| x3+ |
| x5−18 x7+x9 | ⎞ ⎟ ⎟ ⎠ | e |
| ⎤ ⎥ ⎥ ⎦ |
(1+x2)2 y″(x)+3 y(x)=0 |
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ |
| ,− |
|
y″− |
| =x+4. |
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ |
− |
|
yh=C1 y1+C2 y2= |
| , C1,C2∈ℝ. |
yp=−y1 | ∫ |
| dx+y2 | ∫ |
| dx, |
W=y1 y2′−y2 y1′≠ 0. |
W:=y1*y2’-y2*y1’:; f:=x+4:; |
yp:=normal(-y1*int(y2*f/W,x)+y2*int(y1*f/W,x)) |
|
purge(C1,C2):; ysol:=yp+C1*y1+C2*y2:; |
normal(diff(ysol,x,2)-27/(36*(x-1)^2)*ysol)==f |
true |
y″= | ⎛ ⎜ ⎜ ⎝ |
| − |
| − |
| ⎞ ⎟ ⎟ ⎠ | y. |
r:=-3/(16x^2)-2/(9*(x-1)^2)+3/(16x*(x-1)):; |
kovacicsols(y’’=r*y) |
−omega_4*x4*(x−1)4+omega_3*x3*(x−1)3*(7*x−3)/3−omega_2*x2*(x−1)2*(48*x2−41*x+9)/24+omega_*x*(x−1)*(320*x3−409*x2+180*x−27)/432+(−2048*x4+3484*x3−2313*x2+702*x−81)/20736 |
48 t (t+1) (5 t−4) y″+8 (25 t+16) (t−2) y′−(5 t+68) y=0. |
| ω_4 | ⎛ ⎝ | 135 t4−616 t3−144 t2+3072 t−4096 | ⎞ ⎠ | − |
| ω_3 t | ⎛ ⎝ | t+1 | ⎞ ⎠ | ⎛ ⎝ | 23 t2−92 t+128 | ⎞ ⎠ | − |
| ω_2 t2 | ⎛ ⎝ | t+1 | ⎞ ⎠ | ⎛ ⎝ | 15 t3−80 t2+80 t+256 | ⎞ ⎠ | + |
| ω_ t3 | ⎛ ⎝ | t−4 | ⎞ ⎠ | ⎛ ⎝ | t+1 | ⎞ ⎠ | 2 | ⎛ ⎝ | 5 t+8 | ⎞ ⎠ | −t4 | ⎛ ⎝ | t+1 | ⎞ ⎠ | 2 | ⎛ ⎝ | t+4 | ⎞ ⎠ | ⎛ ⎝ | 5 t+4 | ⎞ ⎠ |