78.19.15 problem 4 (d)

Internal problem ID [18434]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 4 (d)
Date solved : Tuesday, January 28, 2025 at 11:49:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 45

Order:=6; 
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);
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 84

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

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