22.5.9 problem 9.9

Internal problem ID [4596]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.9
Date solved : Tuesday, September 30, 2025 at 07:36:41 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+\sin \left (x \right ) y^{\prime }+\cos \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: 39

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

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

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

\[ y(x)\to c_2 \left (\frac {x^5}{10}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {x^4}{6}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.928 (sec). Leaf size: 105

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{4} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{24} + \frac {x^{4} \cos ^{2}{\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{6} - \frac {x^{2} \cos {\left (x \right )}}{2} + 1\right ) + C_{1} x \left (- \frac {x^{3} \sin ^{3}{\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{12} + \frac {x^{2} \sin ^{2}{\left (x \right )}}{6} - \frac {x^{2} \cos {\left (x \right )}}{6} - \frac {x \sin {\left (x \right )}}{2} + 1\right ) + O\left (x^{6}\right ) \]