Internal problem ID [10862]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 38.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.36 (sec). Leaf size: 294
dsolve(diff(y(x),x$2)+(a*b*x^2+b*x+2*a)*diff(y(x),x)+a^2*(b*x^2+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (2 a^{2} b^{2} x^{2}+2 a b \,x^{2} \sqrt {a^{2} b^{2}}+3 a \,b^{2} x +3 b x \sqrt {a^{2} b^{2}}+12 a \sqrt {a^{2} b^{2}}\right )}{12 \sqrt {a^{2} b^{2}}}} \operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, -\frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 3^{\frac {1}{3}}}{4 \left (a^{2} b^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \,b^{2} \left (2 a x +1\right )}{6 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\left (-2 a^{2} b^{2} x^{2}+2 a b \,x^{2} \sqrt {a^{2} b^{2}}-3 a \,b^{2} x +3 b x \sqrt {a^{2} b^{2}}+12 a \sqrt {a^{2} b^{2}}\right ) x}{12 \sqrt {a^{2} b^{2}}}} \operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, \frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 3^{\frac {1}{3}}}{4 \left (a^{2} b^{2}\right )^{\frac {2}{3}}}, -\frac {\left (a x +\frac {1}{2}\right ) 3^{\frac {2}{3}} b^{2} a}{3 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right ) \]
✓ Solution by Mathematica
Time used: 2.136 (sec). Leaf size: 57
DSolve[y''[x]+(a*b*x^2+b*x+2*a)*y'[x]+a^2*(b*x^2+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {e^{-\frac {1}{6} b K[1]^2 (2 a K[1]+3)}}{(a K[1]+1)^2}dK[1]+c_1\right ) \]