58.15.4 problem 4

Internal problem ID [14862]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:55:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+x^{2} y^{\prime }+\left (x -2\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✗ Maple

Order:=6; 
ode:=(x^5+x^4-6*x^3)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+(x-2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 282

ode=(x^5+x^4-6*x^3)*D[y[x],{x,2}]+x^2*D[y[x],x]+(x-2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/6} \left (-\frac {70670717962217 i x^{9/2}}{8463329722368 \sqrt {3}}+\frac {454703707 i x^{7/2}}{544195584 \sqrt {3}}-\frac {287057 i x^{5/2}}{1679616 \sqrt {3}}+\frac {22 i x^{3/2}}{243 \sqrt {3}}+\frac {28128149072197063 x^5}{1523399350026240}-\frac {222818846149 x^4}{156728328192}+\frac {35197783 x^3}{181398528}-\frac {14123 x^2}{279936}+\frac {17 x}{216}-\frac {7 i \sqrt {x}}{6 \sqrt {3}}+1\right )+c_2 e^{\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/6} \left (\frac {70670717962217 i x^{9/2}}{8463329722368 \sqrt {3}}-\frac {454703707 i x^{7/2}}{544195584 \sqrt {3}}+\frac {287057 i x^{5/2}}{1679616 \sqrt {3}}-\frac {22 i x^{3/2}}{243 \sqrt {3}}+\frac {28128149072197063 x^5}{1523399350026240}-\frac {222818846149 x^4}{156728328192}+\frac {35197783 x^3}{181398528}-\frac {14123 x^2}{279936}+\frac {17 x}{216}+\frac {7 i \sqrt {x}}{6 \sqrt {3}}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + (x - 2)*y(x) + (x**5 + x**4 - 6*x**3)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**2*Derivative(y(x), x) + (x - 2)*y(x) + (x**5 + x**4 - 6*x**3)*Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular