52.4.11 problem 19

Internal problem ID [8321]
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
Date solved : Monday, January 27, 2025 at 03:48:55 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+\left (1-\cos \left (x \right )\right ) y^{\prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 64

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);
 

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{80} x^{5}+\frac {1}{180} x^{6}-\frac {5}{4032} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {1}{120} x^{6}+\frac {73}{60480} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 36

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(1-Cos[x])*D[y[x],x]+x^2*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to -\frac {53 x^7}{8640}+\frac {x^5}{48}+\frac {x^4}{6}-\frac {x^3}{3}-2 x+3 \]