68.16.11 problem 11

Internal problem ID [17804]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 11
Date solved : Thursday, October 02, 2025 at 02:28:24 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

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

\[ y = \left (1-x^{2}-\frac {1}{3} x^{3}+\frac {1}{6} x^{4}+\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {5}{24} x^{4}-\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 68

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

\[ y(x)\to c_1 \left (\frac {x^5}{15}+\frac {x^4}{6}-\frac {x^3}{3}-x^2+1\right )+c_2 \left (-\frac {x^5}{120}-\frac {5 x^4}{24}-\frac {x^3}{3}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.312 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{6} - \frac {x^{3}}{3} - x^{2} + 1\right ) + C_{1} x \left (- \frac {5 x^{3}}{24} - \frac {x^{2}}{3} + \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]