1.10 problem 22

Internal problem ID [4789]

Book: A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section: Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.1.2 page 230
Problem number: 22.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+2 x y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).

Solution by Maple

Time used: 0.015 (sec). Leaf size: 34

Order:=6; 
dsolve(diff(y(x),x$2)+2*x*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

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

Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

AsymptoticDSolveValue[y''[x]+2*x*y'[x]+2*y[x]==0,y[x],{x,0,5}]
 

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