50.22.13 problem 2(e)

Internal problem ID [8156]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number : 2(e)
Date solved : Monday, January 27, 2025 at 03:44:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 40

Order:=8; 
dsolve(2*x*diff(y(x),x$2)+(1-x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x -\frac {1}{120} x^{2}-\frac {1}{1680} x^{3}-\frac {1}{24192} x^{4}-\frac {1}{380160} x^{5}-\frac {1}{6589440} x^{6}-\frac {1}{125798400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 111

AsymptoticDSolveValue[2*x*D[y[x],{x,2}]+(1-x)*D[y[x],x]+(2+x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {17333 x^7}{48432384000}-\frac {34817 x^6}{691891200}-\frac {1171 x^5}{4435200}+\frac {121 x^4}{40320}+\frac {37 x^3}{1680}-\frac {3 x^2}{40}-\frac {x}{2}+1\right )+c_2 \left (\frac {4 x^7}{143325}-\frac {x^6}{8400}-\frac {19 x^5}{6300}-\frac {x^4}{840}+\frac {2 x^3}{15}+\frac {x^2}{6}-2 x+1\right ) \]