23.3.517 problem 523

Internal problem ID [6231]
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 : 523
Date solved : Friday, October 03, 2025 at 01:57:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \operatorname {a0} \operatorname {a1} \left (-k +x \right ) y+\left (1-\operatorname {a0} +\operatorname {a1} +\operatorname {a0} \operatorname {a2} -\operatorname {a3} +\left (\operatorname {a2} +\operatorname {a3} \right ) x +\left (1+\operatorname {a0} +\operatorname {a1} \right ) x^{2}\right ) y^{\prime }+\left (1-x \right ) \left (a -x \right ) x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.235 (sec). Leaf size: 393

ode:=a0*a1*(-k+x)*y(x)+(1-a0+a1+a0*a2-a3+(a2+a3)*x+(1+a0+a1)*x^2)*diff(y(x),x)+(1-x)*(a-x)*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: 2.05 (sec). Leaf size: 209

ode=a0*a1*(-k + x)*y[x] + (1 - a0 + a1 + a0*a2 - a3 + (a2 + a3)*x + (1 + a0 + a1)*x^2)*D[y[x],x] + (1 - x)*(a - x)*x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^{-\frac {\text {a0} \text {a2}+\text {a1}+1}{a}} \left (c_2 x^{\frac {a+\text {a0}+\text {a3}}{a}} \text {HeunG}\left [a,\text {a0} \text {a1} k-\frac {(a (\text {a0} (\text {a2}-1)+\text {a1}+\text {a2}+1)+\text {a0} (\text {a2}-1)+\text {a1}-\text {a3}+1) (a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1)}{a^2},\frac {a \text {a0}+a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1}{a},\frac {a \text {a1}+a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1}{a},\frac {2 a-\text {a0} \text {a2}+\text {a0}-\text {a1}+\text {a3}-1}{a},\frac {\text {a0} \text {a2}+2 \text {a1}+\text {a2}+2}{1-a},x\right ]+c_1 x^{\frac {\text {a0} \text {a2}+\text {a1}+1}{a}} \text {HeunG}\left [a,\text {a0} \text {a1} k,\text {a0},\text {a1},\frac {\text {a0} (\text {a2}-1)+\text {a1}-\text {a3}+1}{a},\frac {\text {a0} \text {a2}+2 \text {a1}+\text {a2}+2}{1-a},x\right ]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
k = symbols("k") 
y = Function("y") 
ode = Eq(a0*a1*(-k + x)*y(x) + x*(1 - x)*(a - x)*Derivative(y(x), (x, 2)) + (a0*a2 - a0 + a1 - a3 + x**2*(a0 + a1 + 1) + x*(a2 + a3) + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None