78.19.8 problem 2 (d)

Internal problem ID [18427]
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 (d)
Date solved : Tuesday, January 28, 2025 at 11:49:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} 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: 0.054 (sec). Leaf size: 107

Order:=6; 
dsolve(x^3*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{12 i \sqrt {3}+24} x^{2}+\frac {1}{1440} \frac {-3 i \sqrt {3}-1}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{12 i \sqrt {3}-24} x^{2}+\frac {-3 \sqrt {3}-i}{7200 i+8640 \sqrt {3}} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 194

AsymptoticDSolveValue[x^3*D[y[x],{x,2}]+Sin[x]*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {\left (\frac {1}{36 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}-\frac {1}{120}\right ) x^4}{1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )}+\frac {x^2}{6 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+1\right ) x^{-(-1)^{2/3}}+c_2 \left (\frac {\left (\frac {1}{36 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {1}{120}\right ) x^4}{1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )}+\frac {x^2}{6 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}+1\right ) x^{\sqrt [3]{-1}} \]