23.3.551 problem 557

Internal problem ID [6265]
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 : 557
Date solved : Friday, October 03, 2025 at 01:58:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {a4} \,x^{4}+\operatorname {a2} \,x^{2}+\operatorname {a0} \right ) y-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 83

ode:=(a4*x^4+a2*x^2+a0)*y(x)-2*x*(-x^2+1)*diff(y(x),x)+(-x^2+1)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, \sqrt {-\operatorname {a0} -\operatorname {a4} -\operatorname {a2}}, \frac {\operatorname {a4}}{4}, \frac {1}{4}-\frac {\operatorname {a0}}{4}, x^{2}\right ) c_2 x +\operatorname {HeunC}\left (0, -\frac {1}{2}, \sqrt {-\operatorname {a0} -\operatorname {a4} -\operatorname {a2}}, \frac {\operatorname {a4}}{4}, \frac {1}{4}-\frac {\operatorname {a0}}{4}, x^{2}\right ) c_1 \right ) \left (x^{2}-1\right )^{\frac {\sqrt {-\operatorname {a0} -\operatorname {a4} -\operatorname {a2}}}{2}} \]

✓ Mathematica. Time used: 0.365 (sec). Leaf size: 287

ode=(a0 + a2*x^2 + a4*x^4)*y[x] - 2*x*(1 - x^2)*D[y[x],x] + (1 - x^2)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{i \sqrt {\text {a4}} x} \left (\frac {x+1}{x-1}\right )^{\frac {1}{2} \sqrt {-\text {a0}-\text {a2}-\text {a4}}} \left (c_2 (x-1)^{\sqrt {-\text {a0}-\text {a2}-\text {a4}}} \text {HeunC}\left [-\sqrt {-\text {a0}-\text {a2}-\text {a4}}-2 i \sqrt {\text {a4}} \left (\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1\right )+\text {a0}-\text {a4},-4 i \sqrt {\text {a4}} \left (\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1\right ),\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1,\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1,-4 i \sqrt {\text {a4}},\frac {1-x}{2}\right ]+c_1 \text {HeunC}\left [2 i \sqrt {\text {a4}} \left (\sqrt {-\text {a0}-\text {a2}-\text {a4}}-1\right )-\text {a2}-2 \text {a4},-4 i \sqrt {\text {a4}},1-\sqrt {-\text {a0}-\text {a2}-\text {a4}},\sqrt {-\text {a0}-\text {a2}-\text {a4}}+1,-4 i \sqrt {\text {a4}},\frac {1-x}{2}\right ]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a2 = symbols("a2") 
a4 = symbols("a4") 
y = Function("y") 
ode = Eq(-2*x*(1 - x**2)*Derivative(y(x), x) + (1 - x**2)**2*Derivative(y(x), (x, 2)) + (a0 + a2*x**2 + a4*x**4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False