44.21.4 problem 2(d)

Internal problem ID [9427]
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(d)
Date solved : Tuesday, September 30, 2025 at 06:18:39 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}-x -6\right ) y^{\prime \prime }+\left (5+3 x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 54

Order:=8; 
ode:=(x^2-x-6)*diff(diff(y(x),x),x)+(5+3*x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=3);
 

\[ y = \frac {c_1 \left (1+\frac {4}{25} \left (x -3\right )-\frac {2}{625} \left (x -3\right )^{2}+\frac {4}{15625} \left (x -3\right )^{3}-\frac {11}{390625} \left (x -3\right )^{4}+\frac {176}{48828125} \left (x -3\right )^{5}-\frac {616}{1220703125} \left (x -3\right )^{6}+\frac {2288}{30517578125} \left (x -3\right )^{7}+\operatorname {O}\left (\left (x -3\right )^{8}\right )\right )}{\left (x -3\right )^{{9}/{5}}}+c_2 \left (1-\frac {1}{14} \left (x -3\right )+\frac {1}{133} \left (x -3\right )^{2}-\frac {1}{1064} \left (x -3\right )^{3}+\frac {1}{7714} \left (x -3\right )^{4}-\frac {5}{262276} \left (x -3\right )^{5}+\frac {5}{1704794} \left (x -3\right )^{6}-\frac {5}{10715848} \left (x -3\right )^{7}+\operatorname {O}\left (\left (x -3\right )^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 145

ode=(x^2-x-6)*D[y[x],{x,2}]+(5+3*x)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,7}]
 

\[ y(x)\to c_1 \left (-\frac {5 (x-3)^7}{10715848}+\frac {5 (x-3)^6}{1704794}-\frac {5 (x-3)^5}{262276}+\frac {(x-3)^4}{7714}-\frac {(x-3)^3}{1064}+\frac {1}{133} (x-3)^2+\frac {3-x}{14}+1\right )+\frac {c_2 \left (\frac {2288 (x-3)^7}{30517578125}-\frac {616 (x-3)^6}{1220703125}+\frac {176 (x-3)^5}{48828125}-\frac {11 (x-3)^4}{390625}+\frac {4 (x-3)^3}{15625}-\frac {2}{625} (x-3)^2+\frac {4 (x-3)}{25}+1\right )}{(x-3)^{9/5}} \]

✓ Sympy. Time used: 0.415 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x + 5)*Derivative(y(x), x) + (x**2 - x - 6)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=3,n=8)
 

\[ y{\left (x \right )} = \frac {C_{2}}{\left (x - 3\right )^{\frac {9}{5}}} + C_{1} + O\left (x^{8}\right ) \]