Internal problem ID [4775]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page
300
Problem number: 7.3.101 (d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\sin \relax (x ) y^{\prime \prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 58
Order:=6; dsolve(sin(x)*diff(y(x),x$2)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1+\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{48} x^{3}+\frac {1}{192} x^{4}+\frac {37}{28800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (x +\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}-\frac {1}{6} x^{3}-\frac {5}{192} x^{4}-\frac {257}{28800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 85
AsymptoticDSolveValue[Sin[x]*y''[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{48} x \left (x^3+4 x^2+24 x+48\right ) \log (x)+\frac {1}{64} \left (-3 x^4-16 x^3-80 x^2-64 x+64\right )\right )+c_2 \left (\frac {x^5}{192}+\frac {x^4}{48}+\frac {x^3}{12}+\frac {x^2}{2}+x\right ) \]