46.2.30 problem 30

Internal problem ID [9567]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 06:20:51 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 52

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\frac {1}{518400} x^{6}-\frac {1}{25401600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}-\frac {49}{5184000} x^{6}+\frac {121}{592704000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 153

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

\[ y(x)\to c_1 \left (-\frac {x^7}{25401600}+\frac {x^6}{518400}-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right )+c_2 \left (\frac {121 x^7}{592704000}-\frac {49 x^6}{5184000}+\frac {137 x^5}{432000}-\frac {25 x^4}{3456}+\frac {11 x^3}{108}-\frac {3 x^2}{4}+\left (-\frac {x^7}{25401600}+\frac {x^6}{518400}-\frac {x^5}{14400}+\frac {x^4}{576}-\frac {x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)+2 x\right ) \]

✓ Sympy. Time used: 0.195 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{7}}{25401600} + \frac {x^{6}}{518400} - \frac {x^{5}}{14400} + \frac {x^{4}}{576} - \frac {x^{3}}{36} + \frac {x^{2}}{4} - x + 1\right ) + O\left (x^{8}\right ) \]