Internal problem ID [10913]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 79.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (-2+b \right ) x^{3} y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 69
dsolve(x*diff(y(x),x$2)+(a*b*x^2+b-5)*diff(y(x),x)+2*a^2*(b-2)*x^3*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a \left (b -2\right ) x^{2}}{2}} \operatorname {KummerM}\left (-1+\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )+c_{2} {\mathrm e}^{-\frac {a \left (b -2\right ) x^{2}}{2}} \operatorname {KummerU}\left (-1+\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right ) \]
✓ Solution by Mathematica
Time used: 3.578 (sec). Leaf size: 67
DSolve[x*y''[x]+(a*b*x^2+b-5)*y'[x]+2*a^2*(b-2)*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x^2} \left (a x^2+1\right ) \left (c_2 \int _1^x\frac {e^{-\frac {1}{2} a (b-4) K[1]^2} K[1]^{5-b}}{\left (a K[1]^2+1\right )^2}dK[1]+c_1\right ) \]