Internal problem ID [8982]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1403.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (\frac {1-\mathit {a1} -\mathit {b1}}{x -\mathit {c1}}+\frac {1-\mathit {a2} -\mathit {b2}}{x -\mathit {c2}}+\frac {1-\mathit {a3} -\mathit {b3}}{x -\mathit {c3}}\right ) y^{\prime }+\frac {\left (\frac {\mathit {a1} \mathit {b1} \left (\mathit {c1} -\mathit {c3} \right ) \left (\mathit {c1} -\mathit {c2} \right )}{x -\mathit {c1}}+\frac {\mathit {a2} \mathit {b2} \left (\mathit {c2} -\mathit {c1} \right ) \left (\mathit {c2} -\mathit {c3} \right )}{x -\mathit {c2}}+\frac {\mathit {a3} \mathit {b3} \left (\mathit {c3} -\mathit {c2} \right ) \left (\mathit {c3} -\mathit {c1} \right )}{x -\mathit {c3}}\right ) y}{\left (x -\mathit {c1} \right ) \left (x -\mathit {c2} \right ) \left (x -\mathit {c3} \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.703 (sec). Leaf size: 311
dsolve(diff(diff(y(x),x),x) = -((1-a1-b1)/(x-c1)+(1-a2-b2)/(x-c2)+(1-a3-b3)/(x-c3))*diff(y(x),x)-1/(x-c1)/(x-c2)/(x-c3)*(a1*b1*(c1-c3)*(c1-c2)/(x-c1)+a2*b2*(c2-c1)*(c2-c3)/(x-c2)+a3*b3*(c3-c2)*(c3-c1)/(x-c3))*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \mathit {HG}\left (\frac {\mathit {c1} -\mathit {c3}}{\mathit {c1} -\mathit {c2}}, \frac {\left (\left (-\mathit {a3} -2 \mathit {b1} -\mathit {b2} +2\right ) \mathit {c1} +\left (\mathit {a3} +\mathit {b1} -1\right ) \mathit {c2} +\mathit {c3} \left (\mathit {b1} +\mathit {b2} -1\right )\right ) \mathit {a1} -\left (\mathit {b1} -1\right ) \left (\mathit {a2} +\mathit {b3} \right ) \mathit {c1} +\left (\mathit {b1} \mathit {b3} -\mathit {a2} \mathit {b2} +\mathit {b3} \left (\mathit {a3} -1\right )\right ) \mathit {c2} +\left (\mathit {b1} \mathit {a2} +\left (\mathit {b2} -1\right ) \mathit {a2} -\mathit {a3} \mathit {b3} \right ) \mathit {c3}}{\mathit {c1} -\mathit {c2}}, \mathit {a1} +\mathit {a2} +\mathit {b3} , -\mathit {b1} -\mathit {a3} -\mathit {b2} +2, \mathit {a1} -\mathit {b1} +1, \mathit {a2} -\mathit {b2} +1, \frac {-x +\mathit {c1}}{\mathit {c1} -\mathit {c2}}\right ) \left (x -\mathit {c1} \right )^{\mathit {a1}} \left (x -\mathit {c2} \right )^{\mathit {a2}} \left (x -\mathit {c3} \right )^{\mathit {b3}}+c_{2} \mathit {HG}\left (\frac {\mathit {c1} -\mathit {c3}}{\mathit {c1} -\mathit {c2}}, \frac {\left (\left (-2 \mathit {a1} -\mathit {a3} -\mathit {b2} +2\right ) \mathit {c1} +\left (\mathit {a1} +\mathit {a3} -1\right ) \mathit {c2} +\mathit {c3} \left (\mathit {a1} +\mathit {b2} -1\right )\right ) \mathit {b1} -\left (\mathit {a2} +\mathit {b3} \right ) \left (\mathit {a1} -1\right ) \mathit {c1} +\left (\mathit {a1} \mathit {b3} -\mathit {a2} \mathit {b2} +\mathit {b3} \left (\mathit {a3} -1\right )\right ) \mathit {c2} +\left (\mathit {a1} \mathit {a2} +\left (\mathit {b2} -1\right ) \mathit {a2} -\mathit {a3} \mathit {b3} \right ) \mathit {c3}}{\mathit {c1} -\mathit {c2}}, \mathit {a2} +\mathit {b3} +\mathit {b1} , -\mathit {a3} -\mathit {b2} +2-\mathit {a1} , -\mathit {a1} +\mathit {b1} +1, \mathit {a2} -\mathit {b2} +1, \frac {-x +\mathit {c1}}{\mathit {c1} -\mathit {c2}}\right ) \left (x -\mathit {c1} \right )^{\mathit {b1}} \left (x -\mathit {c2} \right )^{\mathit {a2}} \left (x -\mathit {c3} \right )^{\mathit {b3}} \]
✓ Solution by Mathematica
Time used: 8.023 (sec). Leaf size: 289
DSolve[y''[x] == -((((a1*b1*(c1 - c2)*(c1 - c3))/(-c1 + x) + (a2*b2*(-c1 + c2)*(c2 - c3))/(-c2 + x) + (a3*b3*(-c1 + c3)*(-c2 + c3))/(-c3 + x))*y[x])/((-c1 + x)*(-c2 + x)*(-c3 + x))) - ((1 - a1 - b1)/(-c1 + x) + (1 - a2 - b2)/(-c2 + x) + (1 - a3 - b3)/(-c3 + x))*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (x-\text {c2})^{\text {a2}} (x-\text {c3})^{\text {b3}} \left (c_1 (x-\text {c1})^{\text {a1}} \text {HeunG}\left [\frac {\text {c1}-\text {c3}}{\text {c1}-\text {c2}},\frac {\text {a1} (-(\text {c1} (\text {a3}+2 \text {b1}+\text {b2}-2))+\text {c2} (\text {a3}+\text {b1}-1)+\text {c3} (\text {b1}+\text {b2}-1))+\text {a2} (\text {c3} (\text {b1}+\text {b2}-1)-\text {b1} \text {c1}-\text {b2} \text {c2}+\text {c1})+\text {b3} (\text {c2} (\text {a3}+\text {b1}-1)-\text {a3} \text {c3}-\text {b1} \text {c1}+\text {c1})}{\text {c1}-\text {c2}},-\text {a3}-\text {b1}-\text {b2}+2,\text {a1}+\text {a2}+\text {b3},\text {a1}-\text {b1}+1,\text {a2}-\text {b2}+1,\frac {\text {c1}-x}{\text {c1}-\text {c2}}\right ]+c_2 (x-\text {c1})^{\text {b1}} \text {HeunG}\left [\frac {\text {c1}-\text {c3}}{\text {c1}-\text {c2}},\frac {\text {a2} (\text {c3} (\text {a1}+\text {b2}-1)-\text {a1} \text {c1}-\text {b2} \text {c2}+\text {c1})+\text {b1} (-(\text {c1} (2 \text {a1}+\text {a3}+\text {b2}-2))+\text {c2} (\text {a1}+\text {a3}-1)+\text {c3} (\text {a1}+\text {b2}-1))+\text {b3} (\text {c2} (\text {a1}+\text {a3}-1)-\text {a1} \text {c1}-\text {a3} \text {c3}+\text {c1})}{\text {c1}-\text {c2}},-\text {a1}-\text {a3}-\text {b2}+2,\text {a2}+\text {b1}+\text {b3},-\text {a1}+\text {b1}+1,\text {a2}-\text {b2}+1,\frac {\text {c1}-x}{\text {c1}-\text {c2}}\right ]\right ) \\ \end{align*}