85.74.2 problem 2

Internal problem ID [22964]
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 317
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:17:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 34

Order:=6; 
ode:=cos(x)*diff(diff(y(x),x),x)+y(x)*sin(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: 0.001 (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}]
 

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

✓ Sympy. Time used: 1.293 (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)
 

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