61.28.15 problem 75

Internal problem ID [12575]
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
Problem number : 75
Date solved : Tuesday, January 28, 2025 at 03:22:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (a x +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (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)
 

\[ y = -\pi \,\operatorname {erf}\left (\frac {2 b x +a}{2 \sqrt {-b}}\right ) {\mathrm e}^{-\frac {2 b^{2} x^{2}+a^{2}}{4 b}} c_{2} a -2 \sqrt {\pi }\, \sqrt {-b}\, {\mathrm e}^{\frac {1}{2} b \,x^{2}+a x} c_{2} +c_{1} {\mathrm e}^{-\frac {b \,x^{2}}{2}} \]

✓ Solution by Mathematica

Time used: 0.552 (sec). Leaf size: 46

DSolve[x*D[y[x],{x,2}]-(a*x+1)*D[y[x],x]-b*x^2*(b*x+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {b x^2}{2}} \left (c_2 \int _1^xe^{b K[1]^2+a K[1]+1} K[1]dK[1]+c_1\right ) \]