61.27.17 problem 27

Internal problem ID [12527]
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
Problem number : 27
Date solved : Tuesday, January 28, 2025 at 03:19:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 49

dsolve(diff(y(x),x$2)+(a*x+2*b)*diff(y(x),x)+(a*b*x-a+b^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.254 (sec). Leaf size: 64

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

\[ y(x)\to x e^{-b x} \left (-\sqrt {\frac {\pi }{2}} \sqrt {a} c_2 \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )-\frac {c_2 e^{-\frac {a x^2}{2}}}{x}+c_1\right ) \]