46.1.14 problem 12

Internal problem ID [9517]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:19:58 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*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 54

ode=D[y[x],{x,2}]+2*x*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 0.220 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + 2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

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