20.25.12 problem 13

Internal problem ID [4017]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 13
Date solved : Monday, January 27, 2025 at 08:06:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 32

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

\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{5} x^{2}+\frac {1}{90} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{3} x^{2}+\frac {1}{42} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^4}{90}+\frac {x^2}{5}+1\right )+c_2 \left (\frac {x^4}{42}+\frac {x^2}{3}+1\right ) \]