14.28.12 problem 12

Internal problem ID [4074]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Additional problems. Section 11.7. page 788
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 07:01:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 62

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

\[ y = x \left (c_1 x \left (1-x +\frac {1}{3} x^{2}-\frac {1}{36} x^{3}-\frac {7}{720} x^{4}+\frac {31}{10800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\left (-x +x^{2}-\frac {1}{3} x^{3}+\frac {1}{36} x^{4}+\frac {7}{720} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (1-x -\frac {1}{2} x^{2}+\frac {19}{36} x^{3}-\frac {53}{432} x^{4}-\frac {1}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{36} x^2 \left (x^3-12 x^2+36 x-36\right ) \log (x)-\frac {1}{432} x \left (65 x^4-372 x^3+648 x^2-432\right )\right )+c_2 \left (-\frac {7 x^6}{720}-\frac {x^5}{36}+\frac {x^4}{3}-x^3+x^2\right ) \]

✓ Sympy. Time used: 0.298 (sec). Leaf size: 24

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

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