50.19.4 problem 1(d)

Internal problem ID [8112]
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 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 1(d)
Date solved : Sunday, March 30, 2025 at 12:45:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 72

Order:=8; 
ode:=(3*x+1)*x*diff(diff(y(x),x),x)-(1+x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-2 x +\frac {17}{4} x^{2}-\frac {289}{30} x^{3}+\frac {5491}{240} x^{4}-\frac {236113}{4200} x^{5}+\frac {28569673}{201600} x^{6}-\frac {28569673}{78400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (2 x^{2}-4 x^{3}+\frac {17}{2} x^{4}-\frac {289}{15} x^{5}+\frac {5491}{120} x^{6}-\frac {236113}{2100} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-4 x +6 x^{2}-12 x^{3}+\frac {209}{8} x^{4}-\frac {54247}{900} x^{5}+\frac {521849}{3600} x^{6}-\frac {158526173}{441000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.403 (sec). Leaf size: 118

ode=(3*x+1)*x*D[y[x],{x,2}]-(x+1)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {27353 x^6-12886 x^5+6525 x^4-3600 x^3+1800 x^2+7200 x+3600}{3600}-\frac {1}{240} x^2 \left (5491 x^4-2312 x^3+1020 x^2-480 x+240\right ) \log (x)\right )+c_2 \left (\frac {28569673 x^8}{201600}-\frac {236113 x^7}{4200}+\frac {5491 x^6}{240}-\frac {289 x^5}{30}+\frac {17 x^4}{4}-2 x^3+x^2\right ) \]

✓ Sympy. Time used: 1.036 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(3*x + 1)*Derivative(y(x), (x, 2)) - (x + 1)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {x^{5}}{9450} + \frac {x^{4}}{540} - \frac {x^{3}}{45} + \frac {x^{2}}{6} - \frac {2 x}{3} + 1\right ) + O\left (x^{8}\right ) \]