62.32.10 problem Ex 10

Internal problem ID [12908]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 55. Summary. Page 129
Problem number : Ex 10
Date solved : Friday, March 14, 2025 at 12:17:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }+2 x^{3} \left (1+x \right ) y^{\prime }+n^{2} y&=0 \end{align*}

✓ Maple. Time used: 1.240 (sec). Leaf size: 283

ode:=x^4*diff(diff(y(x),x),x)+2*x^3*(1+x)*diff(y(x),x)+n^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} {\mathrm e}^{\frac {i \sqrt {-n^{2}}\, x^{2}+i n^{2}-n \,x^{2}}{n x}} \operatorname {HeunD}\left (8 \left (-n^{2}\right )^{{1}/{4}}, \frac {-8 i \left (-n^{2}\right )^{{3}/{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{{3}/{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{{3}/{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{{1}/{4}} x -i n}{\left (-n^{2}\right )^{{1}/{4}} x +i n}\right )+c_{2} {\mathrm e}^{\frac {-i \sqrt {-n^{2}}\, x^{2}-i n^{2}-n \,x^{2}}{n x}} \operatorname {HeunD}\left (-8 \left (-n^{2}\right )^{{1}/{4}}, \frac {-8 i \left (-n^{2}\right )^{{3}/{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{{3}/{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{{3}/{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{{1}/{4}} x -i n}{\left (-n^{2}\right )^{{1}/{4}} x +i n}\right )}{\sqrt {x}} \]

✗ Mathematica

ode=x^4*D[y[x],{x,2}]+2*x^3*(1+x)*D[y[x],x]+n^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(n**2*y(x) + x**4*Derivative(y(x), (x, 2)) + 2*x**3*(x + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-n**2*y(x) - x**4*Derivative(y(x), (x, 2)))/(2*x**3*(x + 1)) cannot be solved by the factorable group method