12.14.25 problem 25

Internal problem ID [1966]
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 : 25
Date solved : Tuesday, March 04, 2025 at 01:47:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 38

Order:=6; 
ode:=6*x^2*diff(diff(y(x),x),x)+x*(10-x)*diff(y(x),x)-(x+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1+\frac {2}{21} x +\frac {1}{180} x^{2}+\frac {1}{4212} x^{3}+\frac {1}{124416} x^{4}+\frac {1}{4432320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 53

ode=6*x^2*D[y[x],{x,2}]+x*(10-x)*D[y[x],x]-(2+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {x^5}{4432320}+\frac {x^4}{124416}+\frac {x^3}{4212}+\frac {x^2}{180}+\frac {2 x}{21}+1\right )+\frac {c_2}{x} \]

✓ Sympy. Time used: 0.882 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x**2*Derivative(y(x), (x, 2)) + x*(10 - x)*Derivative(y(x), x) - (x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} \left (\frac {x^{4}}{124416} + \frac {x^{3}}{4212} + \frac {x^{2}}{180} + \frac {2 x}{21} + 1\right ) + \frac {C_{1}}{x} + O\left (x^{6}\right ) \]