84.32.7 problem 19.19

Internal problem ID [22315]
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 : 19.19
Date solved : Thursday, October 02, 2025 at 08:37:25 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-x y^{\prime }&={\mathrm e}^{-x} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 46

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

\[ y = y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{8}-\frac {x^{5}}{30}+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 53

ode=D[y[x],{x,2}]-x*D[y[x],x]==Exp[-x]; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {x^5}{30}+\frac {x^4}{8}-\frac {x^3}{6}+\frac {x^2}{2}+c_2 \left (\frac {x^5}{40}+\frac {x^3}{6}+x\right )+c_1 \]

✗ Sympy

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

ValueError : ODE -x*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-x) does not match hint 2nd_power_series_regular