23.3.40 problem 40

Internal problem ID [5754]
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 : 40
Date solved : Friday, October 03, 2025 at 01:43:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{4}+\operatorname {a1} \,x^{2}+\operatorname {a0} \right ) y+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 109

ode:=(x^4+a1*x^2+a0)*y(x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{\frac {i x \left (2 x^{2}+3 \operatorname {a1} \right )}{6}} \operatorname {HeunT}\left (\frac {\left (\operatorname {a1}^{2}-4 \operatorname {a0} \right ) 3^{{2}/{3}} 2^{{1}/{3}}}{8}, 0, -\frac {\operatorname {a1} 2^{{2}/{3}} 3^{{1}/{3}}}{2}, \frac {i 2^{{1}/{3}} 3^{{2}/{3}} x}{3}\right )+c_2 \,{\mathrm e}^{-\frac {i x \left (2 x^{2}+3 \operatorname {a1} \right )}{6}} \operatorname {HeunT}\left (\frac {\left (\operatorname {a1}^{2}-4 \operatorname {a0} \right ) 3^{{2}/{3}} 2^{{1}/{3}}}{8}, 0, -\frac {\operatorname {a1} 2^{{2}/{3}} 3^{{1}/{3}}}{2}, -\frac {i 2^{{1}/{3}} 3^{{2}/{3}} x}{3}\right ) \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 100

ode=(a0 + a1*x^2 + x^4)*y[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1{}^2 e^{\frac {1}{6} i x \left (3 \text {a1}+2 x^2\right )} \text {HeunT}\left [\frac {\text {a1}^2}{4}-\text {a0},2 i,i \text {a1},0,2 i,x\right ]+c_2 e^{-\frac {1}{6} i x \left (3 \text {a1}+2 x^2\right )} \text {HeunT}\left [\frac {\text {a1}^2}{4}-\text {a0},-2 i,-i \text {a1},0,-2 i,x\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
y = Function("y") 
ode = Eq((a0 + a1*x**2 + x**4)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False