88.25.13 problem 11

Internal problem ID [24218]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 212
Problem number : 11
Date solved : Thursday, October 02, 2025 at 10:00:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+\left (x^{3}+11 x \right ) y^{\prime }+\left (-2 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 35

Order:=6; 
ode:=6*x^2*diff(diff(y(x),x),x)+(x^3+11*x)*diff(y(x),x)+(-2*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {5}{44} x^{2}+\frac {5}{8096} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+\frac {c_2 \left (1+\frac {7}{78} x^{2}+\frac {7}{23400} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 52

ode=6*x^2*D[y[x],{x,2}]+(11*x+x^3)*D[y[x],{x,1}]+(1-2*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (\frac {7 x^4}{23400}+\frac {7 x^2}{78}+1\right )}{\sqrt [3]{x}}+\frac {c_2 \left (\frac {5 x^4}{8096}+\frac {5 x^2}{44}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.398 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x**2*Derivative(y(x), (x, 2)) + (1 - 2*x**2)*y(x) + (x**3 + 11*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2} \left (\frac {7 x^{4}}{23400} + \frac {7 x^{2}}{78} + 1\right )}{\sqrt [3]{x}} + \frac {C_{1} \left (\frac {5 x^{4}}{8096} + \frac {5 x^{2}}{44} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]