54.3.18 problem 18

Internal problem ID [8574]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 18
Date solved : Monday, January 27, 2025 at 04:16:38 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 49

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

\[ y = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 54

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

\[ y(x)\to c_2 \left (-\frac {8 x^7}{105}+\frac {4 x^5}{15}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {x^6}{6}+\frac {x^4}{2}-x^2+1\right ) \]