7.14.30 problem 30

Internal problem ID [455]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 30
Date solved : Monday, January 27, 2025 at 02:53:47 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 49

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

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {7}{360} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {7 x^5}{360}+\frac {x^4}{18}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {x^5}{60}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]