64.15.11 problem 11

Internal problem ID [13509]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 10:02:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 44

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

\[ y = c_{1} x^{{1}/{3}} \left (1+\frac {7}{12} x +\frac {5}{36} x^{2}+\frac {13}{648} x^{3}+\frac {1}{486} x^{4}+\frac {19}{116640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {3}{10} x^{2}+\frac {1}{20} x^{3}+\frac {1}{176} x^{4}+\frac {3}{6160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 81

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {19 x^5}{116640}+\frac {x^4}{486}+\frac {13 x^3}{648}+\frac {5 x^2}{36}+\frac {7 x}{12}+1\right )+c_2 \left (\frac {3 x^5}{6160}+\frac {x^4}{176}+\frac {x^3}{20}+\frac {3 x^2}{10}+x+1\right ) \]

✓ Sympy. Time used: 0.938 (sec). Leaf size: 66

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

\[ y{\left (x \right )} = C_{2} \left (\frac {3 x^{5}}{6160} + \frac {x^{4}}{176} + \frac {x^{3}}{20} + \frac {3 x^{2}}{10} + x + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {x^{4}}{486} + \frac {13 x^{3}}{648} + \frac {5 x^{2}}{36} + \frac {7 x}{12} + 1\right ) + O\left (x^{6}\right ) \]