84.32.4 problem 19.16

Internal problem ID [22312]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 19. Power series solutions about an ordinary point. Supplementary problems
Problem number : 19.16
Date solved : Thursday, October 02, 2025 at 08:37:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -2 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {\left (x +2\right )^{3}}{6}-\frac {\left (x +2\right )^{4}}{6}-\frac {\left (x +2\right )^{5}}{30}\right ) y \left (-2\right )+\left (x +2+2 \left (x +2\right )^{2}+2 \left (x +2\right )^{3}+\frac {2 \left (x +2\right )^{4}}{3}-\frac {17 \left (x +2\right )^{5}}{30}\right ) y^{\prime }\left (-2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 74

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

\[ y(x)\to c_1 \left (-\frac {1}{30} (x+2)^5-\frac {1}{6} (x+2)^4-\frac {1}{6} (x+2)^3+1\right )+c_2 \left (-\frac {17}{30} (x+2)^5+\frac {2}{3} (x+2)^4+2 (x+2)^3+2 (x+2)^2+x+2\right ) \]

✓ Sympy. Time used: 0.348 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = C_{2} \left (\frac {13 \left (x + 2\right )^{5}}{540} + \frac {\left (x + 2\right )^{4}}{27} + \frac {\left (x + 2\right )^{3}}{18} + 1\right ) + C_{1} \left (x + \frac {\left (x + 2\right )^{5}}{45} + \frac {\left (x + 2\right )^{4}}{36} + 2\right ) + O\left (x^{6}\right ) \]