54.9.13 problem 13

Internal problem ID [8683]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 13
Date solved : Monday, January 27, 2025 at 04:19:10 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 2 x y^{\prime \prime }+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: 52

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

\[ y = c_{1} \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\frac {1}{97297200} x^{6}-\frac {1}{10216206000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\frac {1}{7484400} x^{6}-\frac {1}{681080400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^7}{10216206000}+\frac {x^6}{97297200}-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right )+c_2 \left (-\frac {x^7}{681080400}+\frac {x^6}{7484400}-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right ) \]