73.23.24 problem 33.5 (L)

Internal problem ID [15670]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (L)
Date solved : Tuesday, January 28, 2025 at 08:03:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x y^{\prime }-2 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 44

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

\[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 49

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

\[ y(x)\to c_1 \left (\frac {x^5}{20}+\frac {x^3}{3}+1\right )+c_2 \left (\frac {x^5}{40}+\frac {x^4}{6}+\frac {x^3}{6}+x\right ) \]