50.21.3 problem 2(x)

Internal problem ID [8140]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.6. Gauss Hypergeometric Equation. Page 187
Problem number : 2(x)
Date solved : Wednesday, March 05, 2025 at 05:30:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }+\left (5 x +4\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -1 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 54

Order:=8; 
ode:=(x^2-1)*diff(diff(y(x),x),x)+(5*x+4)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=-1);
 

\[ y = c_{1} \sqrt {x +1}\, \left (1+\frac {25}{12} \left (x +1\right )+\frac {245}{96} \left (x +1\right )^{2}+\frac {315}{128} \left (x +1\right )^{3}+\frac {4235}{2048} \left (x +1\right )^{4}+\frac {13013}{8192} \left (x +1\right )^{5}+\frac {75075}{65536} \left (x +1\right )^{6}+\frac {206635}{262144} \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right )+c_{2} \left (1+4 \left (x +1\right )+6 \left (x +1\right )^{2}+\frac {32}{5} \left (x +1\right )^{3}+\frac {40}{7} \left (x +1\right )^{4}+\frac {32}{7} \left (x +1\right )^{5}+\frac {112}{33} \left (x +1\right )^{6}+\frac {1024}{429} \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 139

ode=(x^2-1)*D[y[x],{x,2}]+(5*x+4)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,7}]
 

\[ y(x)\to c_1 \sqrt {x+1} \left (\frac {206635 (x+1)^7}{262144}+\frac {75075 (x+1)^6}{65536}+\frac {13013 (x+1)^5}{8192}+\frac {4235 (x+1)^4}{2048}+\frac {315}{128} (x+1)^3+\frac {245}{96} (x+1)^2+\frac {25 (x+1)}{12}+1\right )+c_2 \left (\frac {1024}{429} (x+1)^7+\frac {112}{33} (x+1)^6+\frac {32}{7} (x+1)^5+\frac {40}{7} (x+1)^4+\frac {32}{5} (x+1)^3+6 (x+1)^2+4 (x+1)+1\right ) \]

✓ Sympy. Time used: 12.515 (sec). Leaf size: 1681

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((5*x + 4)*Derivative(y(x), x) + (x**2 - 1)*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=-1,n=8)
 

\[ \text {Solution too large to show} \]