12.15.26 problem 22

Internal problem ID [2024]
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 : 22
Date solved : Tuesday, March 04, 2025 at 01:48:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6-x \right ) y^{\prime }+\left (8-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:=2*x^2*(1+x)*diff(diff(y(x),x),x)-x*(6-x)*diff(y(x),x)+(8-x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {5}{2} x +\frac {35}{8} x^{2}-\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}-\frac {3003}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {3}{2} x -\frac {57}{16} x^{2}+\frac {583}{96} x^{3}-\frac {13771}{1536} x^{4}+\frac {187339}{15360} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 128

ode=2*x^2*(1+x)*D[y[x],{x,2}]-x*(6-x)*D[y[x],x]+(8-x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {3003 x^5}{256}+\frac {1155 x^4}{128}-\frac {105 x^3}{16}+\frac {35 x^2}{8}-\frac {5 x}{2}+1\right ) x^2+c_2 \left (\left (\frac {187339 x^5}{15360}-\frac {13771 x^4}{1536}+\frac {583 x^3}{96}-\frac {57 x^2}{16}+\frac {3 x}{2}\right ) x^2+\left (-\frac {3003 x^5}{256}+\frac {1155 x^4}{128}-\frac {105 x^3}{16}+\frac {35 x^2}{8}-\frac {5 x}{2}+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 1.036 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x + 1)*Derivative(y(x), (x, 2)) - x*(6 - x)*Derivative(y(x), x) + (8 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{2} + O\left (x^{6}\right ) \]