84.31.4 problem 19.4

Internal problem ID [22302]
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. Solved problems. Page 98
Problem number : 19.4
Date solved : Thursday, October 02, 2025 at 08:37:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 69

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

\[ y = \left (1+\frac {3 \left (x +1\right )^{2}}{2}+\frac {\left (x +1\right )^{3}}{6}+\frac {\left (x +1\right )^{4}}{6}-\frac {7 \left (x +1\right )^{5}}{60}\right ) y \left (-1\right )+\left (x +1+\frac {\left (x +1\right )^{2}}{2}+\frac {\left (x +1\right )^{3}}{2}-\frac {\left (x +1\right )^{5}}{20}\right ) y^{\prime }\left (-1\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (-\frac {7}{60} (x+1)^5+\frac {1}{6} (x+1)^4+\frac {1}{6} (x+1)^3+\frac {3}{2} (x+1)^2+1\right )+c_2 \left (-\frac {1}{20} (x+1)^5+\frac {1}{2} (x+1)^3+\frac {1}{2} (x+1)^2+x+1\right ) \]

✓ Sympy. Time used: 0.227 (sec). Leaf size: 49

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

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