12.14.17 problem 17

Internal problem ID [1958]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 17
Date solved : Tuesday, March 04, 2025 at 01:46:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 45

Order:=6; 
ode:=3*x^2*diff(diff(y(x),x),x)+x*(1+x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{3} x +\frac {1}{18} x^{2}-\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}-\frac {1}{29160} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}}+c_2 x \left (1-\frac {1}{7} x +\frac {1}{70} x^{2}-\frac {1}{910} x^{3}+\frac {1}{14560} x^{4}-\frac {1}{276640} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 86

ode=3*x^2*D[y[x],{x,2}]+x*(1+x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (-\frac {x^5}{276640}+\frac {x^4}{14560}-\frac {x^3}{910}+\frac {x^2}{70}-\frac {x}{7}+1\right )+\frac {c_2 \left (-\frac {x^5}{29160}+\frac {x^4}{1944}-\frac {x^3}{162}+\frac {x^2}{18}-\frac {x}{3}+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 0.868 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + x*(x + 1)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{14560} - \frac {x^{3}}{910} + \frac {x^{2}}{70} - \frac {x}{7} + 1\right ) + \frac {C_{1} \left (- \frac {x^{5}}{29160} + \frac {x^{4}}{1944} - \frac {x^{3}}{162} + \frac {x^{2}}{18} - \frac {x}{3} + 1\right )}{\sqrt [3]{x}} + O\left (x^{6}\right ) \]