Internal problem ID [10871]
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: 37.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 298
dsolve(diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+(alpha*x^2+beta*x+gamma)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {\left (\sqrt {a^{2}}+a \right ) \sqrt {a^{2}}\, x^{3}}{6 a}-\frac {b \left (\sqrt {a^{2}}+a \right ) \sqrt {a^{2}}\, x^{2}}{4 a^{2}}+\frac {\alpha \sqrt {a^{2}}\, x}{a^{2}}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 \gamma \,a^{2}-a b \beta +b^{2} \alpha +2 \alpha ^{2}\right )}{2 \left (a^{2}\right )^{\frac {4}{3}}}, -\frac {3 \left (a^{2}-\beta a +b \alpha \right ) \sqrt {a^{2}}}{a^{3}}, -\frac {3^{\frac {1}{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {a^{2}}-a \right ) \sqrt {a^{2}}\, x^{3}}{6 a}-\frac {b \left (\sqrt {a^{2}}-a \right ) \sqrt {a^{2}}\, x^{2}}{4 a^{2}}-\frac {\alpha \sqrt {a^{2}}\, x}{a^{2}}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 \gamma \,a^{2}-a b \beta +b^{2} \alpha +2 \alpha ^{2}\right )}{2 \left (a^{2}\right )^{\frac {4}{3}}}, \frac {3 \left (a^{2}-\beta a +b \alpha \right ) \sqrt {a^{2}}}{a^{3}}, -\frac {3^{\frac {1}{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+(a*x^2+b*x)*y'[x]+(\[Alpha]*x^2+\[Beta]*x+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved