9.8.30 problem problem 30

Internal problem ID [1095]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 30
Date solved : Saturday, March 29, 2025 at 10:38:46 PM
CAS classification : [_Lienard]

\begin{align*} x 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: 49

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {7}{360} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {7 x^5}{360}+\frac {x^4}{18}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {x^5}{60}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 1.803 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{120} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]