6.11.12 problem 23

Internal problem ID [1851]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 05:20:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (-x^{2}-6 x +1\right ) y^{\prime }+\left (x^{2}+6 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 46

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

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-12 x +\frac {119}{2} x^{2}-\frac {583}{3} x^{3}+\frac {1981}{4} x^{4}-\frac {80287}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (17 x -\frac {471}{4} x^{2}+445 x^{3}-\frac {118285}{96} x^{4}+\frac {702451}{250} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 114

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

\[ y(x)\to c_1 x \left (-\frac {80287 x^5}{75}+\frac {1981 x^4}{4}-\frac {583 x^3}{3}+\frac {119 x^2}{2}-12 x+1\right )+c_2 \left (x \left (\frac {702451 x^5}{250}-\frac {118285 x^4}{96}+445 x^3-\frac {471 x^2}{4}+17 x\right )+x \left (-\frac {80287 x^5}{75}+\frac {1981 x^4}{4}-\frac {583 x^3}{3}+\frac {119 x^2}{2}-12 x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.650 (sec). Leaf size: 8

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

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