Internal problem ID [9095]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1520.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (n +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.641 (sec). Leaf size: 279
dsolve(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,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, -\frac {\operatorname {a1} \,n^{2}+\operatorname {a1} n -\operatorname {a1} -\operatorname {a2} -\operatorname {a3} +b}{4 \left (-\operatorname {a2} +\operatorname {a1} \right )}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )^{2}+c_{2} \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, -\frac {\operatorname {a1} \,n^{2}+\operatorname {a1} n -3 \operatorname {a1} +b}{4 \left (-\operatorname {a2} +\operatorname {a1} \right )}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )^{2} \left (-x +\operatorname {a1} \right )+c_{3} \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, -\frac {\operatorname {a1} \,n^{2}+\operatorname {a1} n -\operatorname {a1} -\operatorname {a2} -\operatorname {a3} +b}{4 \left (-\operatorname {a2} +\operatorname {a1} \right )}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, -\frac {\operatorname {a1} \,n^{2}+\operatorname {a1} n -3 \operatorname {a1} +b}{4 \left (-\operatorname {a2} +\operatorname {a1} \right )}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \sqrt {-x +\operatorname {a1}} \]
✓ Solution by Mathematica
Time used: 2.931 (sec). Leaf size: 418
DSolve[-(n*(1 + n)*y[x]) - 2*(b + (-3 + n + n^2)*x)*y'[x] + (3*a1*a2 + 3*a1*a3 + 3*a2*a3 - 6*(a1 + a2 + a3)*x + 9*x^2)*y''[x] + 2*(-a1 + x)*(-a2 + x)*(-a3 + x)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 \\ \end{align*}