7.15.4 problem 4

Internal problem ID [460]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 4
Date solved : Saturday, March 29, 2025 at 04:54:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=3*x^3*diff(diff(y(x),x),x)+2*x^2*diff(y(x),x)+(-x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.085 (sec). Leaf size: 282

ode=3*x^3*D[y[x],{x,2}]+2*x^2*D[y[x],x]+(1-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/12} \left (\frac {62734403795075135729047 i x^{9/2}}{1135928886651103739904 \sqrt {3}}-\frac {83137610663649865 i x^{7/2}}{15977652003274752 \sqrt {3}}+\frac {49187170129 i x^{5/2}}{55037657088 \sqrt {3}}-\frac {748637 i x^{3/2}}{1990656 \sqrt {3}}-\frac {285399592561410162222647977 x^5}{2290032635488625139646464}+\frac {168706514298618579533 x^4}{18406255107772514304}-\frac {54361598747213 x^3}{47552535724032}+\frac {337107649 x^2}{1146617856}-\frac {385 x}{13824}+\frac {5 i \sqrt {x}}{48 \sqrt {3}}+1\right )+c_2 e^{\frac {2 i}{\sqrt {3} \sqrt {x}}} x^{5/12} \left (-\frac {62734403795075135729047 i x^{9/2}}{1135928886651103739904 \sqrt {3}}+\frac {83137610663649865 i x^{7/2}}{15977652003274752 \sqrt {3}}-\frac {49187170129 i x^{5/2}}{55037657088 \sqrt {3}}+\frac {748637 i x^{3/2}}{1990656 \sqrt {3}}-\frac {285399592561410162222647977 x^5}{2290032635488625139646464}+\frac {168706514298618579533 x^4}{18406255107772514304}-\frac {54361598747213 x^3}{47552535724032}+\frac {337107649 x^2}{1146617856}-\frac {385 x}{13824}-\frac {5 i \sqrt {x}}{48 \sqrt {3}}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**3*Derivative(y(x), (x, 2)) + 2*x**2*Derivative(y(x), x) + (1 - x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE 3*x**3*Derivative(y(x), (x, 2)) + 2*x**2*Derivative(y(x), x) + (1 - x**2)*y(x) does not match hint 2nd_power_series_regular