44.19.7 problem 2(c)

Internal problem ID [9401]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 2(c)
Date solved : Tuesday, September 30, 2025 at 06:18:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 70

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

\[ y = c_1 x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{144} x^{3}-\frac {13}{2880} x^{4}+\frac {29}{86400} x^{5}+\frac {431}{3628800} x^{6}-\frac {4961}{203212800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{144} x^{4}+\frac {13}{2880} x^{5}-\frac {29}{86400} x^{6}-\frac {431}{3628800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {2}{9} x^{3}-\frac {25}{1728} x^{4}-\frac {689}{86400} x^{5}+\frac {263}{162000} x^{6}+\frac {71809}{762048000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 121

ode=x^2*D[y[x],{x,2}]+Sin[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {2539 x^6-16185 x^5-9750 x^4+396000 x^3-1620000 x^2+1296000 x+1296000}{1296000}-\frac {x \left (29 x^5-390 x^4+600 x^3+7200 x^2-43200 x+86400\right ) \log (x)}{86400}\right )+c_2 \left (\frac {431 x^7}{3628800}+\frac {29 x^6}{86400}-\frac {13 x^5}{2880}+\frac {x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 2.246 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{2} x + C_{1} + O\left (x^{8}\right ) \]