78.3.17 problem 5.a

Internal problem ID [21013]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 5.a
Date solved : Thursday, October 02, 2025 at 07:01:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+\frac {3 y^{\prime }}{x +2}+\frac {\left (1-x \right )^{2} y}{x +3}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)+3/(x+2)*diff(y(x),x)+(1-x)^2/(x+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{2}+\frac {23}{108} x^{3}-\frac {449}{2592} x^{4}+\frac {1891}{12960} x^{5}\right ) y \left (0\right )+\left (x -\frac {3}{4} x^{2}+\frac {4}{9} x^{3}-\frac {143}{432} x^{4}+\frac {1097}{4320} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (\frac {1891 x^5}{12960}-\frac {449 x^4}{2592}+\frac {23 x^3}{108}-\frac {x^2}{6}+1\right )+c_2 \left (\frac {1097 x^5}{4320}-\frac {143 x^4}{432}+\frac {4 x^3}{9}-\frac {3 x^2}{4}+x\right ) \]

✓ Sympy. Time used: 0.488 (sec). Leaf size: 3

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

\[ y{\left (x \right )} = O\left (1\right ) \]