15.25.9 problem 8

Internal problem ID [3396]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 04:37:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 9 x^{2} y^{\prime \prime }+\left (2+3 x \right ) y&=x^{4}+x^{2} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 63

Order:=6; 
ode:=9*x^2*diff(diff(y(x),x),x)+(2+3*x)*y(x) = x^4+x^2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{{1}/{3}} \left (1-\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{480} x^{3}+\frac {1}{21120} x^{4}-\frac {1}{1478400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{{2}/{3}} \left (1-\frac {1}{4} x +\frac {1}{56} x^{2}-\frac {1}{1680} x^{3}+\frac {1}{87360} x^{4}-\frac {1}{6988800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{20}-\frac {3}{1120} x +\frac {1129}{123200} x^{2}-\frac {3387}{22422400} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

✓ Mathematica. Time used: 0.184 (sec). Leaf size: 264

ode=9*x^2*D[y[x],{x,2}]+(2+3*x)*y[x]==x^2+x^4; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^5}{1478400}+\frac {x^4}{21120}-\frac {x^3}{480}+\frac {x^2}{20}-\frac {x}{2}+1\right )+x^{2/3} \left (-\frac {x^5}{6988800}+\frac {x^4}{87360}-\frac {x^3}{1680}+\frac {x^2}{56}-\frac {x}{4}+1\right ) \left (\frac {1057 x^{16/3}}{337920}-\frac {241 x^{13/3}}{6240}+\frac {21 x^{10/3}}{200}-\frac {x^{7/3}}{14}+\frac {x^{4/3}}{4}\right )+\sqrt [3]{x} \left (-\frac {x^5}{1478400}+\frac {x^4}{21120}-\frac {x^3}{480}+\frac {x^2}{20}-\frac {x}{2}+1\right ) \left (-\frac {223 x^{17/3}}{212160}+\frac {421 x^{14/3}}{23520}-\frac {57 x^{11/3}}{616}+\frac {x^{8/3}}{32}-\frac {x^{5/3}}{5}\right )+c_2 x^{2/3} \left (-\frac {x^5}{6988800}+\frac {x^4}{87360}-\frac {x^3}{1680}+\frac {x^2}{56}-\frac {x}{4}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 + 9*x**2*Derivative(y(x), (x, 2)) - x**2 + (3*x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE -x**4 + 9*x**2*Derivative(y(x), (x, 2)) - x**2 + (3*x + 2)*y(x) does not match hint 2nd_power_series_regular