36.1.13 problem 3.24 (h)

Internal problem ID [8121]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.24 (h)
Date solved : Tuesday, September 30, 2025 at 05:15:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x \left (x +2\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 38

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

\[ y = c_1 \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {7}{32} x^{2}-\frac {3}{128} x^{3}+\frac {11}{2048} x^{4}-\frac {13}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+4 x +2 x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 60

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

\[ y(x)\to c_2 \left (2 x^2+4 x+1\right )+c_1 \sqrt {x} \left (-\frac {13 x^5}{8192}+\frac {11 x^4}{2048}-\frac {3 x^3}{128}+\frac {7 x^2}{32}+\frac {5 x}{4}+1\right ) \]

✓ Sympy. Time used: 0.435 (sec). Leaf size: 73

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

\[ y{\left (x \right )} = C_{2} \left (\frac {4096 x^{5}}{14175} + \frac {512 x^{4}}{315} + \frac {256 x^{3}}{45} + \frac {32 x^{2}}{3} + 8 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {512 x^{4}}{2835} + \frac {256 x^{3}}{315} + \frac {32 x^{2}}{15} + \frac {8 x}{3} + 1\right ) + O\left (x^{6}\right ) \]