60.4.62 problem 1520
Internal
problem
ID
[11485]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
3,
linear
third
order
Problem
number
:
1520
Date
solved
:
Thursday, March 13, 2025 at 08:53:25 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} 2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y&=0 \end{align*}
✓ Maple. Time used: 7.118 (sec). Leaf size: 288
ode:=2*(x-a1)*(x-a2)*(x-a3)*diff(diff(diff(y(x),x),x),x)+(9*x^2-6*(a1+a2+a3)*x+3*a1*a2+3*a1*a3+3*a2*a3)*diff(diff(y(x),x),x)-2*((n^2+n-3)*x+b)*diff(y(x),x)-n*(n+1)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -c_{2} \left (x -\operatorname {a1} \right ) {\operatorname {HeunG}\left (\frac {-\operatorname {a3} +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +3\right ) \operatorname {a1} -b}{-4 \operatorname {a2} +4 \operatorname {a1}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )}^{2}+c_3 \operatorname {HeunG}\left (\frac {-\operatorname {a3} +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +1\right ) \operatorname {a1} -b +\operatorname {a2} +\operatorname {a3}}{-4 \operatorname {a2} +4 \operatorname {a1}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \operatorname {HeunG}\left (\frac {-\operatorname {a3} +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +3\right ) \operatorname {a1} -b}{-4 \operatorname {a2} +4 \operatorname {a1}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \sqrt {-x +\operatorname {a1}}+c_{1} {\operatorname {HeunG}\left (\frac {-\operatorname {a3} +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +1\right ) \operatorname {a1} -b +\operatorname {a2} +\operatorname {a3}}{-4 \operatorname {a2} +4 \operatorname {a1}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )}^{2}
\]
✓ Mathematica. Time used: 3.185 (sec). Leaf size: 418
ode=-(n*(1 + n)*y[x]) - 2*(b + (-3 + n + n^2)*x)*D[y[x],x] + (3*a1*a2 + 3*a1*a3 + 3*a2*a3 - 6*(a1 + a2 + a3)*x + 9*x^2)*D[y[x],{x,2}] + 2*(-a1 + x)*(-a2 + x)*(-a3 + x)*Derivative[3][y][x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {c_3 (\text {a1}-x) \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},-\frac {\text {a1} \left (n^2+n-3\right )+b}{4 (\text {a1}-\text {a2})},\frac {3}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+3\right ),\frac {3}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]^2}{\text {a1}-\text {a2}}+c_2 \sqrt {\frac {\text {a1}-x}{\text {a1}-\text {a2}}} \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},\frac {-\text {a1} \left (n^2+n-1\right )+\text {a2}+\text {a3}-b}{4 (\text {a1}-\text {a2})},\frac {1}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+1\right ),\frac {1}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ] \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},-\frac {\text {a1} \left (n^2+n-3\right )+b}{4 (\text {a1}-\text {a2})},\frac {3}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+3\right ),\frac {3}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]+c_1 \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},\frac {-\text {a1} \left (n^2+n-1\right )+\text {a2}+\text {a3}-b}{4 (\text {a1}-\text {a2})},\frac {1}{4}-\frac {1}{4} \sqrt {(2 n+1)^2},\frac {1}{4} \left (\sqrt {(2 n+1)^2}+1\right ),\frac {1}{2},\frac {1}{2},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]^2
\]
✗ Sympy
from sympy import *
x = symbols("x")
a1 = symbols("a1")
a2 = symbols("a2")
a3 = symbols("a3")
b = symbols("b")
n = symbols("n")
y = Function("y")
ode = Eq(-n*(n + 1)*y(x) + (-2*a1 + 2*x)*(-a2 + x)*(-a3 + x)*Derivative(y(x), (x, 3)) - (2*b + 2*x*(n**2 + n - 3))*Derivative(y(x), x) + (3*a1*a2 + 3*a1*a3 + 3*a2*a3 + 9*x**2 - x*(6*a1 + 6*a2 + 6*a3))*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-a1*a2*a3*Derivative(y(x), (x, 3)) + a1*a2*x*Derivative(y(x), (x, 3)) + 3*a1*a2*Derivative(y(x), (x, 2))/2 + a1*a3*x*Derivative(y(x), (x, 3)) + 3*a1*a3*Derivative(y(x), (x, 2))/2 - a1*x**2*Derivative(y(x), (x, 3)) - 3*a1*x*Derivative(y(x), (x, 2)) + a2*a3*x*Derivative(y(x), (x, 3)) + 3*a2*a3*Derivative(y(x), (x, 2))/2 - a2*x**2*Derivative(y(x), (x, 3)) - 3*a2*x*Derivative(y(x), (x, 2)) - a3*x**2*Derivative(y(x), (x, 3)) - 3*a3*x*Derivative(y(x), (x, 2)) - n**2*y(x)/2 - n*y(x)/2 + x**3*Derivative(y(x), (x, 3)) + 9*x**2*Derivative(y(x), (x, 2))/2)/(b + n**2*x + n*x - 3*x) cannot be solved by the factorable group method