23.3.477 problem 483

Internal problem ID [6191]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 483
Date solved : Tuesday, September 30, 2025 at 02:36:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \operatorname {a2} y+\operatorname {a1} \left (b x +a \right ) y^{\prime }+\left (b x +a \right )^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 81

ode:=a2*y(x)+a1*(b*x+a)*diff(y(x),x)+(b*x+a)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (\frac {b x +a}{b}\right )^{\frac {-\operatorname {a1} +b +\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} b +b^{2}-4 \operatorname {a2}}}{2 b}}+c_2 \left (\frac {b x +a}{b}\right )^{-\frac {\operatorname {a1} -b +\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} b +b^{2}-4 \operatorname {a2}}}{2 b}} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 95

ode=a2*y[x] + a1*(a + b*x)*D[y[x],x] + (a + b*x)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (a+b x)^{-\frac {\sqrt {\text {a2}} \sqrt {\frac {\text {a1}^2-2 \text {a1} b-4 \text {a2}+b^2}{\text {a2}}}+\text {a1}-b}{2 b}} \left (c_2 (a+b x)^{\frac {\sqrt {\text {a2}} \sqrt {\frac {\text {a1}^2-2 \text {a1} b-4 \text {a2}+b^2}{\text {a2}}}}{b}}+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a1*(a + b*x)*Derivative(y(x), x) + a2*y(x) + (a + b*x)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False