78.19.7 problem 2 (c)

Internal problem ID [18426]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 2 (c)
Date solved : Tuesday, January 28, 2025 at 11:49:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 58

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{144} x^{3}-\frac {13}{2880} x^{4}+\frac {29}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{144} x^{4}+\frac {13}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {2}{9} x^{3}-\frac {25}{1728} x^{4}-\frac {689}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 85

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+Sin[x]*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {-13 x^4+528 x^3-2160 x^2+1728 x+1728}{1728}-\frac {1}{144} x \left (x^3+12 x^2-72 x+144\right ) \log (x)\right )+c_2 \left (-\frac {13 x^5}{2880}+\frac {x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]