12.15.21 problem 17

Internal problem ID [2019]
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 : 17
Date solved : Tuesday, March 04, 2025 at 01:48:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 48

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

\[ y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} x -\frac {1}{32} x^{2}+\frac {1}{128} x^{3}-\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {3}{4} x +\frac {3}{64} x^{2}-\frac {7}{768} x^{3}+\frac {61}{24576} x^{4}-\frac {391}{491520} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 134

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {7 x^5}{8192}-\frac {5 x^4}{2048}+\frac {x^3}{128}-\frac {x^2}{32}+\frac {x}{4}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {391 x^5}{491520}+\frac {61 x^4}{24576}-\frac {7 x^3}{768}+\frac {3 x^2}{64}-\frac {3 x}{4}\right )+\sqrt {x} \left (\frac {7 x^5}{8192}-\frac {5 x^4}{2048}+\frac {x^3}{128}-\frac {x^2}{32}+\frac {x}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 1.000 (sec). Leaf size: 12

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

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