87.25.3 problem 3

Internal problem ID [23806]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 232
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:45:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -2 \end{align*}

With initial conditions

\begin{align*} y \left (-2\right )&=1 \\ y^{\prime }\left (-2\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 13

Order:=6; 
ode:=diff(diff(y(x),x),x)-2*(x+2)*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(-2) = 1, D(y)(-2) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=-2);
 

\[ y = 1-2 \left (x +2\right )^{2} \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 12

ode=D[y[x],{x,2}]-2*(x+2)*D[y[x],x]+4*y[x]==0; 
ic={y[-2]==1,Derivative[1][y][-2] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-2,5}]
 

\[ y(x)\to 1-2 (x+2)^2 \]

✓ Sympy. Time used: 0.198 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(2*x + 4)*Derivative(y(x), x) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(-2): 1, Subs(Derivative(y(x), x), x, -2): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-2,n=6)
 

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