72.19.14 problem 4 (c)

Internal problem ID [19719]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 4 (c)
Date solved : Thursday, October 02, 2025 at 04:41:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {7}{6} x +\frac {21}{40} x^{2}-\frac {11}{80} x^{3}+\frac {143}{5760} x^{4}-\frac {13}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-3 x +2 x^{2}-\frac {2}{3} x^{3}+\frac {1}{7} x^{4}-\frac {1}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 81

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {13 x^5}{3840}+\frac {143 x^4}{5760}-\frac {11 x^3}{80}+\frac {21 x^2}{40}-\frac {7 x}{6}+1\right )+c_2 \left (-\frac {x^5}{45}+\frac {x^4}{7}-\frac {2 x^3}{3}+2 x^2-3 x+1\right ) \]

✓ Sympy. Time used: 0.320 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{5}}{45} + \frac {x^{4}}{7} - \frac {2 x^{3}}{3} + 2 x^{2} - 3 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {143 x^{4}}{5760} - \frac {11 x^{3}}{80} + \frac {21 x^{2}}{40} - \frac {7 x}{6} + 1\right ) + O\left (x^{6}\right ) \]