36.6.25 problem 28

Internal problem ID [6386]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 28
Date solved : Monday, January 27, 2025 at 01:58:38 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 49

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

\[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}-\frac {x^{4}}{24}+\frac {x^{5}}{40}+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 56

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

\[ y(x)\to \frac {x^5}{40}-\frac {x^4}{24}+c_2 \left (\frac {x^4}{12}+x\right )+\frac {x^2}{2}+c_1 \left (-\frac {x^5}{120}+\frac {x^3}{6}+1\right ) \]