Internal problem ID [15473]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 18.1 Integration of differential equation in
series. Power series. Exercises page 171
Problem number: 729.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x y^{\prime \prime }+\sin \left (x \right ) y=x} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 1, y^{\prime }\left (\pi \right ) = 0] \end {align*}
With the expansion point for the power series method at \(x = \pi \).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 35
Order:=6; dsolve([x*diff(y(x),x$2)+y(x)*sin(x)=x,y(Pi) = 1, D(y)(Pi) = 0],y(x),type='series',x=Pi);
\[ y \left (x \right ) = 1+\frac {1}{2} \left (-\pi +x \right )^{2}+\frac {1}{6} \frac {1}{\pi } \left (-\pi +x \right )^{3}-\frac {1}{12} \frac {1}{\pi ^{2}} \left (-\pi +x \right )^{4}+\frac {1}{60} \frac {\pi ^{2}+3}{\pi ^{3}} \left (-\pi +x \right )^{5}+\operatorname {O}\left (\left (-\pi +x \right )^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 75
AsymptoticDSolveValue[{x*y''[x]+Sin[x]*y[x]==x,{y[Pi]==1,y'[Pi]==0}},y[x],{x,Pi,5}]
\[ y(x)\to \frac {1}{60} \left (\frac {3}{2 \pi }-\frac {\pi ^2-6}{2 \pi ^3}\right ) (x-\pi )^5-\frac {(x-\pi )^4}{12 \pi ^2}+\frac {(x-\pi )^3}{6 \pi }+\frac {1}{2} (x-\pi )^2+1 \]