Internal problem ID [10865]
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: 31.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 317
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+(alpha*x^2+beta*x+gamma)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} \operatorname {hypergeom}\left (\left [\frac {\left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+a^{3}-2 \gamma \,a^{2}+\left (2 b \beta -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (\left (a x +2 b \right ) \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+\left (a^{2}-4 \alpha \right ) \left (a^{2} x +2 a b -4 \alpha x -4 \beta \right )\right ) x}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}}+c_{2} \left (a^{2} x +a b -4 \alpha x -2 \beta \right ) \operatorname {hypergeom}\left (\left [\frac {3 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+a^{3}-2 \gamma \,a^{2}+\left (2 b \beta -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (\left (a x +2 b \right ) \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+\left (a^{2}-4 \alpha \right ) \left (a^{2} x +2 a b -4 \alpha x -4 \beta \right )\right ) x}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}} \]
✓ Solution by Mathematica
Time used: 0.467 (sec). Leaf size: 307
DSolve[y''[x]+(a*x+b)*y'[x]+(\[Alpha]*x^2+\[Beta]*x+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (-\frac {x \left (2 b \sqrt {a^2-4 \alpha }+a \left (x \sqrt {a^2-4 \alpha }+2 b\right )+a^2 x-4 (\beta +\alpha x)\right )}{4 \sqrt {a^2-4 \alpha }}\right ) \left (c_1 \operatorname {HermiteH}\left (\frac {-a^3-\left (\sqrt {a^2-4 \alpha }-2 \gamma \right ) a^2+(4 \alpha -2 b \beta ) a+2 \left (\alpha b^2+\beta ^2+2 \sqrt {a^2-4 \alpha } \alpha -4 \alpha \gamma \right )}{2 \left (a^2-4 \alpha \right )^{3/2}},\frac {x a^2+b a-2 (2 x \alpha +\beta )}{\sqrt {2} \left (a^2-4 \alpha \right )^{3/4}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3+\left (\sqrt {a^2-4 \alpha }-2 \gamma \right ) a^2+(2 b \beta -4 \alpha ) a-2 \left (\alpha b^2+\beta ^2+2 \sqrt {a^2-4 \alpha } \alpha -4 \alpha \gamma \right )}{4 \left (a^2-4 \alpha \right )^{3/2}},\frac {1}{2},\frac {\left (x a^2+b a-2 (2 x \alpha +\beta )\right )^2}{2 \left (a^2-4 \alpha \right )^{3/2}}\right )\right ) \]