Internal problem ID [10984]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 150.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+y^{\prime } b -6 y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 75
dsolve((x^2-a^2)*diff(y(x),x$2)+b*diff(y(x),x)-6*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\frac {b^{3}}{24}+\frac {b^{2} x}{4}+\frac {\left (-5 a^{2}+9 x^{2}\right ) b}{12}-a^{2} x +x^{3}\right )+c_{2} \left (a -x \right ) \left (x +a \right ) \left (b -4 x \right ) \left (\frac {x +a}{a -x}\right )^{\frac {b}{2 a}} \]
✓ Solution by Mathematica
Time used: 13.059 (sec). Leaf size: 1171
DSolve[(x^2-a^2)*y''[x]+b*y'[x]-6*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {b \text {arctanh}\left (\frac {x}{a}\right )}{2 a}+\frac {\left (b^5-20 a^2 b^3+64 a^4 b+\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}\right ) \text {RootSum}\left [-b^3-6 \text {$\#$1} b^2+10 a^2 b-18 \text {$\#$1}^2 b-24 \text {$\#$1}^3+24 a^2 \text {$\#$1}\&,\log (x-\text {$\#$1})\&\right ]}{2 \left (b^5-20 a^2 b^3+64 a^4 b\right )}} (a+x)^{\frac {1}{2}-\frac {\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}}{4 a b \left (32 a^3-16 b a^2-2 b^2 a+b^3\right )}} (4 x-b)^{\frac {b^5}{2 \left (b^5-20 a^2 b^3+64 a^4 b\right )}-\frac {10 a^2 b^3}{b^5-20 a^2 b^3+64 a^4 b}+\frac {32 a^4 b}{b^5-20 a^2 b^3+64 a^4 b}-\frac {\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}}{2 \left (b^5-20 a^2 b^3+64 a^4 b\right )}} c_2 \int _1^x-\frac {e^{-\frac {\left (b^5-20 a^2 b^3+64 a^4 b+\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}\right ) \text {RootSum}\left [-b^3-6 \text {$\#$1} b^2+10 a^2 b-18 \text {$\#$1}^2 b-24 \text {$\#$1}^3+24 a^2 \text {$\#$1}\&,\log (K[1]-\text {$\#$1})\&\right ]}{b^5-20 a^2 b^3+64 a^4 b}} (K[1]-a)^{\frac {\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}}{2 a (4 a-b) b (2 a+b) (4 a+b)}} (a+K[1])^{\frac {\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}}{2 a b \left (32 a^3-16 b a^2-2 b^2 a+b^3\right )}-1} (4 K[1]-b)^{-\frac {b^5}{b^5-20 a^2 b^3+64 a^4 b}+\frac {20 a^2 b^3}{b^5-20 a^2 b^3+64 a^4 b}-\frac {64 a^4 b}{b^5-20 a^2 b^3+64 a^4 b}+\frac {\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}}{b^5-20 a^2 b^3+64 a^4 b}}}{a-K[1]}dK[1] (x-a)^{\frac {1}{2}-\frac {\sqrt {b^2 \left (64 a^4-20 b^2 a^2+b^4\right )^2}}{4 a (4 a-b) b (2 a+b) (4 a+b)}}+e^{\frac {1}{2} \left (\frac {b \text {arctanh}\left (\frac {x}{a}\right )}{a}+\frac {\left (b^5-20 a^2 b^3+64 a^4 b+\sqrt {\left (b^5-20 a^2 b^3+64 a^4 b\right )^2}\right ) \text {RootSum}\left [-b^3-6 \text {$\#$1} b^2+10 a^2 b-18 \text {$\#$1}^2 b-24 \text {$\#$1}^3+24 a^2 \text {$\#$1}\&,\log (x-\text {$\#$1})\&\right ]}{b^5-20 a^2 b^3+64 a^4 b}\right )} (a+x)^{\frac {1}{4} \left (2-\frac {\sqrt {\left (b^5-20 a^2 b^3+64 a^4 b\right )^2}}{a b \left (32 a^3-16 b a^2-2 b^2 a+b^3\right )}\right )} (4 x-b)^{\frac {b^5-20 a^2 b^3+64 a^4 b-\sqrt {\left (b^5-20 a^2 b^3+64 a^4 b\right )^2}}{2 \left (b^5-20 a^2 b^3+64 a^4 b\right )}} c_1 (x-a)^{\frac {1}{2}-\frac {\sqrt {\left (b^5-20 a^2 b^3+64 a^4 b\right )^2}}{4 a (4 a-b) b (2 a+b) (4 a+b)}} \]