78.18.4 problem 3

Internal problem ID [18345]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 28. Second Order Linear Equations. Ordinary Points. Problems at page 217
Problem number : 3
Date solved : Monday, March 31, 2025 at 05:26:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (\frac {x^5}{120}-\frac {x^4}{24}+\frac {x^3}{6}+1\right )+c_2 \left (-\frac {x^5}{30}+\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.731 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{4}}{24} + \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {x^{3}}{24} + \frac {x^{2}}{6} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]