67.24.27 problem 34.9 b(iii)

Internal problem ID [16994]
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 : 34.9 b(iii)
Date solved : Thursday, October 02, 2025 at 01:41:38 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

Order:=7; 
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);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}-\frac {31}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{7}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

ode=D[y[x],{x,2}]+Sin[x]*D[y[x],x]+Cos[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,6}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{10}-\frac {x^3}{3}+x\right )+c_1 \left (-\frac {31 x^6}{720}+\frac {x^4}{6}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 1.272 (sec). Leaf size: 167

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=7)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5} \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{120} - \frac {x^{5} \sin {\left (x \right )} \cos ^{2}{\left (x \right )}}{60} - \frac {x^{4} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{24} + \frac {x^{4} \cos ^{2}{\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{6} - \frac {x^{2} \cos {\left (x \right )}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4} \sin ^{4}{\left (x \right )}}{120} - \frac {x^{4} \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{40} + \frac {x^{4} \cos ^{2}{\left (x \right )}}{120} - \frac {x^{3} \sin ^{3}{\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )} \cos {\left (x \right )}}{12} + \frac {x^{2} \sin ^{2}{\left (x \right )}}{6} - \frac {x^{2} \cos {\left (x \right )}}{6} - \frac {x \sin {\left (x \right )}}{2} + 1\right ) + O\left (x^{7}\right ) \]