46.4.9 problem 17

Internal problem ID [9605]
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 : 17
Date solved : Tuesday, September 30, 2025 at 06:21:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 64

Order:=8; 
ode:=(1-2*sin(x))*diff(diff(y(x),x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{5} x^{5}-\frac {1}{4} x^{6}-\frac {85}{252} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}-\frac {1}{10} x^{5}-\frac {2}{15} x^{6}-\frac {13}{72} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 77

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

\[ y(x)\to c_2 \left (-\frac {13 x^7}{72}-\frac {2 x^6}{15}-\frac {x^5}{10}-\frac {x^4}{12}+x\right )+c_1 \left (-\frac {85 x^7}{252}-\frac {x^6}{4}-\frac {x^5}{5}-\frac {x^4}{6}-\frac {x^3}{6}+1\right ) \]

✓ Sympy. Time used: 0.947 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{6}}{180 \left (4 \sin ^{2}{\left (x \right )} - 4 \sin {\left (x \right )} + 1\right )} + \frac {x^{3}}{6 \left (2 \sin {\left (x \right )} - 1\right )} + 1\right ) + C_{1} x \left (\frac {x^{6}}{504 \left (4 \sin ^{2}{\left (x \right )} - 4 \sin {\left (x \right )} + 1\right )} + \frac {x^{3}}{12 \left (2 \sin {\left (x \right )} - 1\right )} + 1\right ) + O\left (x^{8}\right ) \]