43.2.15 problem 7.3.101 (e)

Internal problem ID [6872]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.101 (e)
Date solved : Sunday, March 30, 2025 at 11:25:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \cos \left (x \right ) y^{\prime \prime }-\sin \left (x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 34

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

\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 35

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

\[ y(x)\to c_2 \left (\frac {x^4}{12}+x\right )+c_1 \left (\frac {x^5}{60}+\frac {x^3}{6}+1\right ) \]

✓ Sympy. Time used: 3.384 (sec). Leaf size: 41

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

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