61.27.14 problem 24

Internal problem ID [12524]
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 : 24
Date solved : Tuesday, January 28, 2025 at 03:19:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 65

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

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

✓ Solution by Mathematica

Time used: 0.411 (sec). Leaf size: 68

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

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