1.15.26 problem 26

Internal problem ID [482]
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 : 26
Date solved : Tuesday, September 30, 2025 at 03:59:18 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x y^{\prime \prime }+\left (-2 x^{2}+1\right ) y^{\prime }-4 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 32

Order:=6; 
ode:=2*x*diff(diff(y(x),x),x)+(-2*x^2+1)*diff(y(x),x)-4*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {2}{3} x^{2}+\frac {4}{21} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {4 x^4}{21}+\frac {2 x^2}{3}+1\right ) \]

✓ Sympy. Time used: 0.395 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*y(x) + 2*x*Derivative(y(x), (x, 2)) + (1 - 2*x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {4 x^{4}}{21} + \frac {2 x^{2}}{3} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{8} + \frac {x^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]