73.24.17 problem 34.7 (c)

Internal problem ID [15697]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.7 (c)
Date solved : Tuesday, January 28, 2025 at 08:05:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-3 x 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: 39

Order:=4; 
dsolve(x*diff(y(x),x$2)-3*x*diff(y(x),x)+sin(x)*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 42

AsymptoticDSolveValue[x*D[y[x],{x,2}]-3*x*D[y[x],x]+Sin[x]*y[x]==0,y[x],{x,0,"4"-1}]
 

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