85.79.4 problem 9

Internal problem ID [22993]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. C Exercises at page 342
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:17:21 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 58

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

\[ y = c_1 x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}-\frac {1}{144} x^{3}+\frac {1}{2880} x^{4}-\frac {1}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{144} x^{4}-\frac {1}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {7}{36} x^{3}-\frac {35}{1728} x^{4}+\frac {101}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{144} x \left (x^3-12 x^2+72 x-144\right ) \log (x)+\frac {-47 x^4+480 x^3-2160 x^2+1728 x+1728}{1728}\right )+c_2 \left (\frac {x^5}{2880}-\frac {x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.189 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{2880} - \frac {x^{3}}{144} + \frac {x^{2}}{12} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]