44.18.10 problem 4(b)

Internal problem ID [9390]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 4(b)
Date solved : Tuesday, September 30, 2025 at 06:18:08 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*}

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

Order:=8; 
ode:=diff(diff(y(x),x),x)+diff(y(x),x)-x*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}+\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 47

ode=D[y[x],{x,2}]+D[y[x],x]-x*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to -\frac {x^7}{1680}+\frac {x^6}{90}-\frac {x^5}{30}+\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 58

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 = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{240} + \frac {x^{5}}{120} - \frac {x^{4}}{24} + \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {x^{5}}{90} - \frac {x^{4}}{30} + \frac {x^{3}}{24} + \frac {x^{2}}{6} - \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]