Internal problem ID [9664]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1330.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}+\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}=0} \]
✓ Solution by Maple
Time used: 0.516 (sec). Leaf size: 1147
dsolve(diff(diff(y(x),x),x) = -(A*x^2+B*x+C)/(x-a)/(x-b)/(x-c)*diff(y(x),x)-(DD*x+E)/(x-a)/(x-b)/(x-c)*y(x),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 10.917 (sec). Leaf size: 1166
DSolve[y''[x] == -(((E + DD*x)*y[x])/((-a + x)*(-b + x)*(-c + x))) - ((C + B*x + A*x^2)*y'[x])/((-a + x)*(-b + x)*(-c + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (x-a)^{-\frac {C}{(a-b) (a-c)}} \left (c_2 \text {HeunG}\left [\frac {a-c}{a-b},\frac {A^2 b a^4+B^2 a^3+A \left (b^2-a b+(a+b) B+2 C\right ) a^3+(a-b)^2 (a \text {DD}+e) a^2-(a-b) (2 a-b) C a-B \left (a^3-b a^2-3 C a+b C\right ) a+(2 a-b) C^2+c \left (A^2 (a-2 b) a^3-b B^2 a+A \left (-a^3+(4 b+B) a^2-3 b (b+B) a-2 b C\right ) a-2 (a-b)^2 (a \text {DD}+e) a-C^2+(a+b) B \left (a^2-b a-C\right )+\left (3 a^2-4 b a+b^2\right ) C\right )+(a-b) c^2 \left ((A+\text {DD}) a^2+(-2 A b-\text {DD} b+e) a-C-b (B+e)\right )}{(a-b)^3 (a-c)^2},\frac {-\left ((A-1) a^2\right )-(A b+b+2 B+A c+c) a+A b c+b c-2 C+(a-b) (a-c) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 (a-b) (a-c)},\frac {-\left ((b-c) \left (A^2-3 A-4 \text {DD}+2\right ) a^3\right )+\left (\left (A^2-A-4 \text {DD}+2\right ) b^2+2 B b+c \left (A^2-3 A-4 \text {DD}+2\right ) b-(A+1) B c-A C+C+c^2 \left (-3 A^2+5 A+8 \text {DD}-4\right )\right ) a^2+\left (-2 (A+2 \text {DD}-1) c^3+(-3 A B+B-2 b (A+2 \text {DD}-1)) c^2+\left (-2 \left (A^2-A-4 \text {DD}+2\right ) b^2-(A+3) B b-2 B^2-(A+3) C\right ) c-((A-1) b+2 B) C\right ) a-2 C^2+((A-1) b-2 B) c C+c^2 \left (\left (A^2-A-4 \text {DD}+2\right ) b^2+(A+1) B b-2 (A-1) C\right )+2 b c^3 (A+2 \text {DD}-1)+(a-c) \left (-\left ((A-2) (b-c) a^2\right )-\left ((A+2) b^2+2 B b-2 c^2-B c+C\right ) a+(A+2) b^2 c+b \left (-2 c^2+B c-C\right )+2 c C\right ) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 (a-b) (a-c) \left ((A-1) c^2+(a+b+B) c-a b+C+(a-c) (b-c) \sqrt {A^2-2 A-4 \text {DD}+1}\right )},-\frac {(A-2) a^2+(2 b+B+2 c) a-2 b c+C}{(a-b) (a-c)},-\frac {A b^2+B b+C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ] (x-a)^{-\frac {(A-1) a^2+(b+B+c) a-b c}{(a-b) (a-c)}}+c_1 \text {HeunG}\left [\frac {a-c}{a-b},\frac {a \text {DD}+e}{a-b},\frac {1}{2} \left (A+\sqrt {A^2-2 A-4 \text {DD}+1}-1\right ),\frac {4 \text {DD} c^2-B c+b \left (A^2-A-4 \text {DD}\right ) c-C+A (B c+C)-a (b-c) \left (A^2-A-4 \text {DD}\right )+(a A (b-c)-A b c-B c-C) \sqrt {A^2-2 A-4 \text {DD}+1}}{2 \left (A c^2-c^2+b c+B c+a (c-b)+C+(a-c) (b-c) \sqrt {A^2-2 A-4 \text {DD}+1}\right )},\frac {A a^2+B a+C}{(a-b) (a-c)},-\frac {A b^2+B b+C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ] (x-a)^{\frac {C}{(a-b) (a-c)}}\right ) \]