50.22.16 problem 2(h)

Internal problem ID [8159]
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. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number : 2(h)
Date solved : Sunday, March 30, 2025 at 12:47:01 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 52

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\frac {1}{720} x^{6}-\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {11}{36} x^{3}-\frac {25}{288} x^{4}+\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}+\frac {121}{235200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 151

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

\[ y(x)\to c_1 \left (-\frac {x^7}{5040}+\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )+c_2 \left (\frac {121 x^7}{235200}-\frac {49 x^6}{14400}+\frac {137 x^5}{7200}-\frac {25 x^4}{288}+\frac {11 x^3}{36}-\frac {3 x^2}{4}+\left (-\frac {x^7}{5040}+\frac {x^6}{720}-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) \log (x)+x\right ) \]

✓ Sympy. Time used: 0.771 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{7}}{5040} + \frac {x^{6}}{720} - \frac {x^{5}}{120} + \frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right ) + O\left (x^{8}\right ) \]