36.5.7 problem 7

Internal problem ID [6351]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 7
Date solved : Monday, January 27, 2025 at 01:57:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.104 (sec). Leaf size: 58

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

\[ y = c_1 x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {1}{48} x^{3}-\frac {3}{320} x^{4}+\frac {19}{9600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{48} x^{4}+\frac {3}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {1}{4} x^{3}-\frac {5}{576} x^{4}-\frac {437}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{576} \left (7 x^4+192 x^3-720 x^2+576 x+576\right )-\frac {1}{48} x \left (x^3+4 x^2-24 x+48\right ) \log (x)\right )+c_2 \left (-\frac {3 x^5}{320}+\frac {x^4}{48}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]