28.5.9 problem 9.9

Internal problem ID [4596]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.9
Date solved : Monday, January 27, 2025 at 09:25:54 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

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

\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (\frac {x^5}{10}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {x^4}{6}-\frac {x^2}{2}+1\right ) \]