34.4.10 problem 10

Internal problem ID [6064]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 12:10:57 AM
CAS classification : [[_elliptic, _class_II]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 32

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}-\frac {3}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {1}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (-\frac {3 x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {x^4}{128}+\frac {x^2}{4}+\left (-\frac {3 x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.963 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{14400} + \frac {x^{4}}{576} + \frac {x^{3}}{36} + \frac {x^{2}}{4} + x + 1\right ) + O\left (x^{6}\right ) \]