48.9.21 problem 22

Internal problem ID [9977]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 06:45:29 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (-x^{2}+1\right ) y^{\prime \prime }+5 \left (-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: 0.029 (sec). Leaf size: 32

Order:=8; 
ode:=x*(-x^2+1)*diff(diff(y(x),x),x)+5*(-x^2+1)*diff(y(x),x)-4*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1+\frac {1}{3} x^{2}+\frac {1}{6} x^{4}+\frac {1}{10} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (-144+144 x^{2}+\operatorname {O}\left (x^{8}\right )\right )}{x^{4}} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 42

ode=x*(1-x^2)*D[y[x],{x,2}]+5*(1-x^2)*D[y[x],x]-4*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^4}-\frac {1}{x^2}\right )+c_2 \left (\frac {x^6}{10}+\frac {x^4}{6}+\frac {x^2}{3}+1\right ) \]

✓ Sympy. Time used: 0.383 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x**2)*Derivative(y(x), (x, 2)) - 4*x*y(x) + (5 - 5*x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {32 x^{7}}{16372125} + \frac {8 x^{6}}{212625} - \frac {8 x^{5}}{14175} + \frac {2 x^{4}}{315} - \frac {16 x^{3}}{315} + \frac {4 x^{2}}{15} - \frac {4 x}{5} + 1\right ) + O\left (x^{8}\right ) \]