88.26.7 problem 7

Internal problem ID [24227]
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 220
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:01:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x^{4} y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 42

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

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {2}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (8 x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-8 x^{3}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 45

ode=x^2*D[y[x],{x,2}]+(2*x^4)*D[y[x],{x,1}]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x^2-\frac {2 x^5}{9}\right )+c_1 \left (\frac {2}{3} x^2 \log (x)-\frac {8 x^3-9}{9 x}\right ) \]

✓ Sympy. Time used: 0.234 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (1 - \frac {2 x^{3}}{9}\right ) + O\left (x^{6}\right ) \]