54.4.12 problem 12

Internal problem ID [8596]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 12
Date solved : Wednesday, March 05, 2025 at 06:10:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 53

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

\[ y = \frac {c_{1} \left (1-x -\frac {1}{4} x^{2}-\frac {1}{60} x^{3}-\frac {1}{1920} x^{4}-\frac {1}{105600} x^{5}-\frac {1}{8870400} x^{6}-\frac {1}{1055577600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{{1}/{3}}}+c_{2} x \left (1+\frac {1}{7} x +\frac {1}{140} x^{2}+\frac {1}{5460} x^{3}+\frac {1}{349440} x^{4}+\frac {1}{33196800} x^{5}+\frac {1}{4381977600} x^{6}+\frac {1}{766846080000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 112

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

\[ y(x)\to c_1 x \left (\frac {x^7}{766846080000}+\frac {x^6}{4381977600}+\frac {x^5}{33196800}+\frac {x^4}{349440}+\frac {x^3}{5460}+\frac {x^2}{140}+\frac {x}{7}+1\right )+\frac {c_2 \left (-\frac {x^7}{1055577600}-\frac {x^6}{8870400}-\frac {x^5}{105600}-\frac {x^4}{1920}-\frac {x^3}{60}-\frac {x^2}{4}-x+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 0.980 (sec). Leaf size: 80

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{6}}{4381977600} + \frac {x^{5}}{33196800} + \frac {x^{4}}{349440} + \frac {x^{3}}{5460} + \frac {x^{2}}{140} + \frac {x}{7} + 1\right ) + \frac {C_{1} \left (- \frac {x^{7}}{1055577600} - \frac {x^{6}}{8870400} - \frac {x^{5}}{105600} - \frac {x^{4}}{1920} - \frac {x^{3}}{60} - \frac {x^{2}}{4} - x + 1\right )}{\sqrt [3]{x}} + O\left (x^{8}\right ) \]