85.79.1 problem 4

Internal problem ID [22990]
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 : 4
Date solved : Thursday, October 02, 2025 at 09:17:19 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

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

\[ v = \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}\right ) v \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{120} x^{5}\right ) v^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

\[ v(x)\to c_2 \left (\frac {x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^4}{24}-\frac {x^2}{2}+1\right ) \]

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

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

\[ v{\left (x \right )} = C_{2} \left (\frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{6}\right ) + O\left (x^{6}\right ) \]