61.30.21 problem 169

Internal problem ID [12669]
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
Problem number : 169
Date solved : Tuesday, January 28, 2025 at 08:10:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -b \lambda \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.908 (sec). Leaf size: 937

dsolve((a*x^2+b)*diff(y(x),x$2)+(c*x^2+d)*diff(y(x),x)+lambda*((c-a*lambda)*x^2+d-b*lambda)*y(x)=0,y(x), singsol=all)
 

\[ y = \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 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} \left (c_{2} \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}} \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, -\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}, \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 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {\lambda d}{a}-\frac {b c \lambda }{a^{2}}+\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 b^{2} d 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^{{3}/{2}}}\right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )+\left (-2 a \lambda +c \right ) \sqrt {-a b}-2 a x \left (a \lambda -c \right )\right ) b}{8 a^{2} b}}+c_{1} \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}} {\mathrm e}^{\lambda x +\frac {\sqrt {-a b}\, \lambda }{a}-\frac {c x}{a}-\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^{{3}/{2}}}} \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, \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}, \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 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {\lambda d}{a}-\frac {b c \lambda }{a^{2}}+\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 )\right ) \]

✓ Solution by Mathematica

Time used: 1.368 (sec). Leaf size: 64

DSolve[(a*x^2+b)*D[y[x],{x,2}]+(c*x^2+d)*D[y[x],x]+\[Lambda]*((c-a*\[Lambda])*x^2+d-b*\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\lambda (-x)} \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {(c-2 a \lambda ) K[1]^2+d-2 b \lambda }{a K[1]^2+b}dK[1]\right )dK[2]+c_1\right ) \]