67.24.13 problem 34.6 (c)

Internal problem ID [16980]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.6 (c)
Date solved : Thursday, October 02, 2025 at 01:41:28 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 32

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

\[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{8} x^{4}+\frac {1}{15} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to c_1 \left (\frac {x^5}{15}-\frac {x^4}{8}+\frac {x^2}{2}-x+1\right ) \]

✓ Sympy. Time used: 0.280 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*cos(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} - C_{1} x + \frac {C_{1} x^{2}}{2} - \frac {C_{1} x^{4}}{8} + \frac {C_{1} x^{5}}{15} + O\left (x^{6}\right ) \]