39.4.8 problem 16

Internal problem ID [8576]
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. Chapter 6 review exercises. page 253
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 05:39:21 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x +2\right ) y^{\prime \prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

Order:=6; 
ode:=(x+2)*diff(diff(y(x),x),x)+3*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = x -\frac {1}{4} x^{3}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 19

ode=(x+2)*D[y[x],{x,2}]+3*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {x^4}{16}-\frac {x^3}{4}+x \]

✓ Sympy. Time used: 0.236 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x + 2)*Derivative(y(x), (x, 2)) + 3*y(x),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{16} + \frac {x^{3}}{8} - \frac {3 x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{3}}{16} - \frac {x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]