52.2.14 problem 14

Internal problem ID [8265]
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 : 14
Date solved : Sunday, March 30, 2025 at 12:50:20 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+10 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 52

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+diff(y(x),x)+10*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-10 x +25 x^{2}-\frac {250}{9} x^{3}+\frac {625}{36} x^{4}-\frac {125}{18} x^{5}+\frac {625}{324} x^{6}-\frac {3125}{7938} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (20 x -75 x^{2}+\frac {2750}{27} x^{3}-\frac {15625}{216} x^{4}+\frac {3425}{108} x^{5}-\frac {6125}{648} x^{6}+\frac {75625}{37044} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 147

ode=x*D[y[x],{x,2}]+D[y[x],x]+10*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {3125 x^7}{7938}+\frac {625 x^6}{324}-\frac {125 x^5}{18}+\frac {625 x^4}{36}-\frac {250 x^3}{9}+25 x^2-10 x+1\right )+c_2 \left (\frac {75625 x^7}{37044}-\frac {6125 x^6}{648}+\frac {3425 x^5}{108}-\frac {15625 x^4}{216}+\frac {2750 x^3}{27}-75 x^2+\left (-\frac {3125 x^7}{7938}+\frac {625 x^6}{324}-\frac {125 x^5}{18}+\frac {625 x^4}{36}-\frac {250 x^3}{9}+25 x^2-10 x+1\right ) \log (x)+20 x\right ) \]

✓ Sympy. Time used: 0.792 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 10*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {3125 x^{7}}{7938} + \frac {625 x^{6}}{324} - \frac {125 x^{5}}{18} + \frac {625 x^{4}}{36} - \frac {250 x^{3}}{9} + 25 x^{2} - 10 x + 1\right ) + O\left (x^{8}\right ) \]