50.19.17 problem 4(d)

Internal problem ID [8125]
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. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 4(d)
Date solved : Monday, January 27, 2025 at 03:44:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 53

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

\[ y = \frac {c_{1} \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}-\frac {1}{680400} x^{6}-\frac {1}{52390800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\frac {1}{486486000} x^{6}+\frac {1}{57891834000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 112

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

\[ y(x)\to c_1 x \left (\frac {x^7}{57891834000}+\frac {x^6}{486486000}+\frac {x^5}{5405400}+\frac {x^4}{83160}+\frac {x^3}{1890}+\frac {x^2}{70}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {x^7}{52390800}-\frac {x^6}{680400}-\frac {x^5}{12600}-\frac {x^4}{360}-\frac {x^3}{18}-\frac {x^2}{2}-x+1\right )}{\sqrt {x}} \]