Internal problem ID [10910]
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: 76.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+\left (b \,x^{3}+a^{2} x +a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(x*diff(y(x),x$2)-(2*a*x+1)*diff(y(x),x)+(b*x^3+a^2*x+a)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}}+c_{2} {\mathrm e}^{a x -\frac {x^{2} \sqrt {-b}}{2}} \]
✓ Solution by Mathematica
Time used: 0.276 (sec). Leaf size: 59
DSolve[x*y''[x]-(2*a*x+1)*y'[x]+(b*x^3+a^2*x+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{a x-\frac {1}{2} i \sqrt {b} x^2} \left (2 c_1-\frac {i c_2 e^{i \sqrt {b} x^2}}{\sqrt {b}}\right ) \]