Internal problem ID [4534]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-y \sin \left (x \right )-\cos \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
Order:=6; dsolve(diff(y(x),x$2)-sin(x)*y(x)=cos(x),y(x),type='series',x=0);
\[ y \left (x \right ) = \left (1+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}-\frac {x^{4}}{24}+\frac {x^{5}}{40}+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 56
AsymptoticDSolveValue[y''[x]-Sin[x]*y[x]==Cos[x],y[x],{x,0,5}]
\[ y(x)\to \frac {x^5}{40}-\frac {x^4}{24}+c_2 \left (\frac {x^4}{12}+x\right )+\frac {x^2}{2}+c_1 \left (-\frac {x^5}{120}+\frac {x^3}{6}+1\right ) \]