Internal problem ID [10909]
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: 75.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }-\left (a x +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 73
dsolve(x*diff(y(x),x$2)-(a*x+1)*diff(y(x),x)-b*x^2*(b*x+a)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {x^{2} b}{2}}+c_{2} \left (a \pi \,\operatorname {erf}\left (\frac {2 b x +a}{2 \sqrt {-b}}\right ) {\mathrm e}^{-\frac {2 b^{2} x^{2}+a^{2}}{4 b}}+2 \sqrt {-b}\, \sqrt {\pi }\, {\mathrm e}^{\frac {1}{2} x^{2} b +a x}\right ) \]
✓ Solution by Mathematica
Time used: 0.444 (sec). Leaf size: 88
DSolve[x*y''[x]-(a*x+1)*y'[x]-b*x^2*(b*x+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {b x^2}{2}} \left (2 \sqrt {b} \left (c_2 e^{x (a+b x)}+2 b c_1\right )-\sqrt {\pi } a c_2 e^{-\frac {a^2}{4 b}} \text {erfi}\left (\frac {a+2 b x}{2 \sqrt {b}}\right )\right )}{4 b^{3/2}} \]