43.2.14 problem 7.3.101 (d)

Internal problem ID [6871]
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)
Date solved : Monday, January 27, 2025 at 02:32:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 58

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

\[ y = c_1 x \left (1+\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{48} x^{3}+\frac {1}{192} x^{4}+\frac {37}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x +\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}-\frac {1}{6} x^{3}-\frac {5}{192} x^{4}-\frac {257}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{48} x \left (x^3+4 x^2+24 x+48\right ) \log (x)+\frac {1}{64} \left (-3 x^4-16 x^3-80 x^2-64 x+64\right )\right )+c_2 \left (\frac {x^5}{192}+\frac {x^4}{48}+\frac {x^3}{12}+\frac {x^2}{2}+x\right ) \]