7.15.1 problem 1

Internal problem ID [457]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 1
Date solved : Monday, January 27, 2025 at 02:53:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (-x^{3}+x \right ) y^{\prime }+y \sin \left (x \right )&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

Order:=6; 
dsolve(x*diff(y(x),x$2)+(x-x^3)*diff(y(x),x)+sin(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{72} x^{4}-\frac {11}{180} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 63

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(x-x^3)*D[y[x],x]+Sin[x]*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (-\frac {x^5}{15}+\frac {x^4}{8}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {11 x^5}{180}+\frac {x^4}{72}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]