52.4.11 problem 19

Internal problem ID [8321]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW. Page 271
Problem number : 19
Date solved : Sunday, March 30, 2025 at 12:51:54 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+\left (1-\cos \left (x \right )\right ) y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 20

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+(1-cos(x))*diff(y(x),x)+x^2*y(x) = 0; 
ic:=y(0) = 3, D(y)(0) = -2; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 3-2 x -\frac {1}{3} x^{3}+\frac {1}{6} x^{4}+\frac {1}{48} x^{5}-\frac {53}{8640} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 36

ode=x*D[y[x],{x,2}]+(1-Cos[x])*D[y[x],x]+x^2*y[x]==0; 
ic={y[0]==3,Derivative[1][y][0] ==-2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to -\frac {53 x^7}{8640}+\frac {x^5}{48}+\frac {x^4}{6}-\frac {x^3}{3}-2 x+3 \]

✓ Sympy. Time used: 1.235 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + x*Derivative(y(x), (x, 2)) + (1 - cos(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{180} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (\frac {x^{6}}{504} - \frac {x^{3}}{12} + 1\right ) + O\left (x^{8}\right ) \]