73.23.32 problem 33.11 (f)

Internal problem ID [15607]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.11 (f)
Date solved : Monday, March 31, 2025 at 01:42:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 34

Order:=5; 
ode:=diff(diff(y(x),x),x)-sin(x)*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 35

ode=D[y[x],{x,2}]-Sin[x]*D[y[x],x]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
 

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

✓ Sympy. Time used: 1.208 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(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=5)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {x^{2} \sin ^{2}{\left (x \right )}}{6} + \frac {x \sin {\left (x \right )}}{2} + 1\right ) + O\left (x^{5}\right ) \]