12.16.16 problem 12

Internal problem ID [2078]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 12
Date solved : Saturday, March 29, 2025 at 11:47:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1-2 x \right ) y^{\prime }-\left (4+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 58

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(1-2*x)*diff(y(x),x)-(4+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1+x +\frac {7}{12} x^{2}+\frac {1}{4} x^{3}+\frac {11}{128} x^{4}+\frac {143}{5760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (9 x^{4}+9 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-144 x -36 x^{2}+12 x^{3}-\frac {36}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 75

ode=x^2*D[y[x],{x,2}]+x*(1-2*x)*D[y[x],x]-(4+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {3 x^4-16 x^3+48 x^2+192 x+192}{192 x^2}-\frac {1}{16} x^2 \log (x)\right )+c_2 \left (\frac {11 x^6}{128}+\frac {x^5}{4}+\frac {7 x^4}{12}+x^3+x^2\right ) \]

✓ Sympy. Time used: 0.869 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(1 - 2*x)*Derivative(y(x), x) - (x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {x^{3}}{4} + \frac {7 x^{2}}{12} + x + 1\right ) + O\left (x^{6}\right ) \]