Internal problem ID [5695]
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(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{3} y^{\prime \prime }+\sin \relax (x ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.171 (sec). Leaf size: 427
Order:=8; dsolve(x^3*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{12 i \sqrt {3}-24} x^{2}+\frac {1}{1440} \frac {3 i \sqrt {3}-1}{\left (i \sqrt {3}-4\right ) \left (-2+i \sqrt {3}\right )} x^{4}+\frac {1}{362880} \frac {9 i \sqrt {3}+115}{\left (i \sqrt {3}-6\right ) \left (i \sqrt {3}-4\right ) \left (-2+i \sqrt {3}\right )} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{12 i \sqrt {3}+24} x^{2}+\frac {1}{1440} \frac {-3 i \sqrt {3}-1}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}+\frac {1}{362880} \frac {9 i \sqrt {3}-115}{\left (i \sqrt {3}+6\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 410
AsymptoticDSolveValue[x^3*y''[x]+Sin[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {\left (\frac {1}{5040}-\frac {1}{720 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+\frac {\frac {1}{36 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}-\frac {1}{120}}{6 \left (1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )\right )}\right ) x^6}{1+\left (5-(-1)^{2/3}\right ) \left (6-(-1)^{2/3}\right )}+\frac {\left (\frac {1}{36 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}-\frac {1}{120}\right ) x^4}{1+\left (3-(-1)^{2/3}\right ) \left (4-(-1)^{2/3}\right )}+\frac {x^2}{6 \left (1+\left (1-(-1)^{2/3}\right ) \left (2-(-1)^{2/3}\right )\right )}+1\right ) x^{-(-1)^{2/3}}+c_2 \left (\frac {\left (\frac {1}{5040}-\frac {1}{720 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}+\frac {\frac {1}{36 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {1}{120}}{6 \left (1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )\right )}\right ) x^6}{1+\left (5+\sqrt [3]{-1}\right ) \left (6+\sqrt [3]{-1}\right )}+\frac {\left (\frac {1}{36 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}-\frac {1}{120}\right ) x^4}{1+\left (3+\sqrt [3]{-1}\right ) \left (4+\sqrt [3]{-1}\right )}+\frac {x^2}{6 \left (1+\left (1+\sqrt [3]{-1}\right ) \left (2+\sqrt [3]{-1}\right )\right )}+1\right ) x^{\sqrt [3]{-1}} \]