84.31.5 problem 19.5

Internal problem ID [22303]
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 : 19.5
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*}

With initial conditions

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 44

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

\[ y(x)\to -\frac {2}{15} (x+1)^5+\frac {1}{3} (x+1)^4-\frac {2}{3} (x+1)^3+2 (x+1)^2-2 (x+1)+2 \]

✓ Sympy. Time used: 0.228 (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 = {y(-1): 2, Subs(Derivative(y(x), x), x, -1): -2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-1,n=6)
 

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