Internal problem ID [5900]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW.
Page 271
Problem number: 19.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+\left (1-\cos \relax (x )\right ) y^{\prime }+x^{2} y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 59
Order:=8; dsolve(x*diff(y(x),x$2)+(1-cos(x))*diff(y(x),x)+x^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{80} x^{5}+\frac {1}{180} x^{6}-\frac {5}{4032} x^{7}\right ) y \relax (0)+\left (x -\frac {1}{12} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {1}{120} x^{6}+\frac {73}{60480} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 36
AsymptoticDSolveValue[x*y''[x]+(1-Cos[x])*y'[x]+x^2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to -\frac {53 x^7}{8640}+\frac {x^5}{48}+\frac {x^4}{6}-\frac {x^3}{3}-2 x+3 \]