6.11.11 problem 22

Internal problem ID [1850]
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 : 22
Date solved : Tuesday, September 30, 2025 at 05:20:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (4+6 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (576 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+192 x -288 x^{2}+576 x^{3}-576 x^{4}-576 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 48

ode=x^2*(1+x)*D[y[x],{x,2}]+x*(1+2*x)*D[y[x],x]-(4+6*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x^2+c_1 \left (\frac {3 x^4-12 x^3+6 x^2-4 x+3}{3 x^2}-4 x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.534 (sec). Leaf size: 15

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

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