23.3.422 problem 427

Internal problem ID [6136]
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 : 427
Date solved : Friday, October 03, 2025 at 01:46:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \operatorname {a2} y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x \left (\operatorname {a0} +x \right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 230

ode:=a2*y(x)+(b1*x+a1)*diff(y(x),x)+x*(a0+x)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.158 (sec). Leaf size: 156

ode=a2*y[x] + (a1 + b1*x)*D[y[x],x] + x*(a0 + x)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \text {a0}^{\frac {\text {a1}}{\text {a0}}-1} x^{1-\frac {\text {a1}}{\text {a0}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-\frac {\text {a1}}{\text {a0}},\frac {1}{2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-\frac {\text {a1}}{\text {a0}},2-\frac {\text {a1}}{\text {a0}},-\frac {x}{\text {a0}}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {\text {a1}}{\text {a0}},-\frac {x}{\text {a0}}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b1 = symbols("b1") 
y = Function("y") 
ode = Eq(a2*y(x) + x*(a0 + x)*Derivative(y(x), (x, 2)) + (a1 + b1*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None