55.28.15 problem 25

Internal problem ID [13798]
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 : 25
Date solved : Thursday, October 02, 2025 at 08:06:52 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {erf}\left (\frac {\left (a x +b \right ) \sqrt {2}}{2 \sqrt {-a}}\right ) c_1 +c_2 \right ) {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to \frac {e^{-\frac {(a x+b)^2}{2 a}} \left (2 \sqrt {a} c_2 e^{\frac {b^2}{2 a}}+\sqrt {2 \pi } c_1 \text {erfi}\left (\frac {a x+b}{\sqrt {2} \sqrt {a}}\right )\right )}{2 \sqrt {a}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x) + (a*x + b)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False