12.15.25 problem 21

Internal problem ID [2023]
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 II. Exercises 7.6. Page 374
Problem number : 21
Date solved : Tuesday, March 04, 2025 at 01:48:20 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (3+5 x \right ) y^{\prime }+\left (1-2 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(2*x+1)*diff(diff(y(x),x),x)+x*(3+5*x)*diff(y(x),x)+(1-2*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+3 x +\frac {3}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{8} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-5\right ) x -\frac {25}{4} x^{2}+\frac {5}{4} x^{3}-\frac {25}{32} x^{4}+\frac {113}{160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 122

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

\[ y(x)\to \frac {c_1 \left (-\frac {3 x^5}{8}+\frac {3 x^4}{8}-\frac {x^3}{2}+\frac {3 x^2}{2}+3 x+1\right )}{x}+c_2 \left (\frac {\frac {113 x^5}{160}-\frac {25 x^4}{32}+\frac {5 x^3}{4}-\frac {25 x^2}{4}-5 x}{x}+\frac {\left (-\frac {3 x^5}{8}+\frac {3 x^4}{8}-\frac {x^3}{2}+\frac {3 x^2}{2}+3 x+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 1.079 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) + x*(5*x + 3)*Derivative(y(x), x) + (1 - 2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x} + O\left (x^{6}\right ) \]