Internal problem ID [8909]
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]]
Solve \begin {gather*} \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 (\mathit {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.123 (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: 2.754 (sec). Leaf size: 666
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]
\begin{align*} y(x)\to c_2 (x-a)^{-\frac {a^2 (A-1)+a (b+B+c)-b c+C}{(a-b) (a-c)}} \text {HeunG}\left [\frac {a-c}{a-b},\frac {a^5 \text {DD}+a^4 (A ((A-1) b+(A-1) c+B)-2 \text {DD} (b+c)-B+e)+a^3 \left (-2 A^2 b c+A \left (c (4 b+B)+b (b+B)+c^2+2 C\right )+b^2 \text {DD}+b B+4 b c \text {DD}-2 e (b+c)+B^2+B c+c^2 \text {DD}-2 C\right )+a^2 \left (-3 A b c (b+B+c)+b^2 (e-2 c \text {DD})+b \left (-2 c^2 \text {DD}+4 e c+3 C\right )+3 C (B+c)+e c^2\right )-a \left (b^2 (c (-c (2 A+\text {DD})+B+2 e)+C)+b \left (2 c ((A+2) C+e c)+B^2 c+B \left (c^2+C\right )\right )+C (c (B+c)-2 C)\right )+e b^2 c^2+(b c-C) (b B c+C (b+c))}{(a-b)^3 (a-c)^2},\frac {-\left (a^2 (A-1)\right )-a (A b+A c+b+2 B+c)+(a-b) (a-c) \sqrt {(A-1)^2-4 \text {DD}}+A b c+b c-2 C}{2 (a-b) (a-c)},\frac {1}{2} \left (\frac {-a (A (a+b)+2 B)+A c (b-a)-2 C}{(a-b) (a-c)}-\sqrt {(A-1)^2-4 \text {DD}}+1\right ),-\frac {a (a (A-2)+2 b+B)+2 c (a-b)+C}{(a-b) (a-c)},-\frac {b (A b+B)+C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ]+c_1 \text {HeunG}\left [\frac {a-c}{a-b},\frac {a \text {DD}+e}{a-b},\frac {1}{2} \left (\sqrt {(A-1)^2-4 \text {DD}}+A-1\right ),\frac {1}{2} \left (-\sqrt {(A-1)^2-4 \text {DD}}+A-1\right ),\frac {a (a A+B)+C}{(a-b) (a-c)},-\frac {b (A b+B)+C}{(a-b) (b-c)},\frac {a-x}{a-b}\right ] \\ \end{align*}