Internal problem ID [11054]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 229.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right ) y^{\prime }+2 \left (-a d +b c \right ) x y=0} \]
✓ Solution by Maple
Time used: 0.312 (sec). Leaf size: 866
dsolve((a*x^2+b)^2*diff(y(x),x$2)+(a*x^2+b)*(c*x^2+d)*diff(y(x),x)+2*(b*c-a*d)*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (-a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}} \left (c_{1} \left (a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} {\mathrm e}^{\frac {\sqrt {-a b}\, c}{2 a^{2}}-\frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d}{2 \sqrt {a}\, \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c}{2 a^{\frac {3}{2}}}} \operatorname {HeunC}\left (\frac {2 \sqrt {-\frac {b}{a}}\, c}{a}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, 0, \frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right )+c_{2} \left (a x +\sqrt {-a b}\right )^{-\frac {-2 a^{2} b +\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} \operatorname {HeunC}\left (\frac {2 \sqrt {-\frac {b}{a}}\, c}{a}, -\frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, 0, \frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) {\mathrm e}^{\frac {i \pi \sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}-i \pi \sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-4 \left (a^{2} \left (\frac {d}{\sqrt {b}\, \sqrt {a}}-\frac {\sqrt {b}\, c}{a^{\frac {3}{2}}}\right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )-\sqrt {-a b}\, c \right ) b}{8 a^{2} b}}\right ) \]
✓ Solution by Mathematica
Time used: 0.165 (sec). Leaf size: 104
DSolve[(a*x^2+b)^2*y''[x]+(a*x^2+b)*(c*x^2+d)*y'[x]+2*(b*c-a*d)*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right ) (b c-a d)}{a^{3/2} \sqrt {b}}-\frac {c x}{a}\right ) \left (\int _1^x\exp \left (\frac {(a d-b c) \arctan \left (\frac {\sqrt {a} K[1]}{\sqrt {b}}\right )}{a^{3/2} \sqrt {b}}+\frac {c K[1]}{a}\right ) c_1dK[1]+c_2\right ) \]