56.4.40 problem 37

Internal problem ID [8929]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 37
Date solved : Sunday, March 30, 2025 at 01:55:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (3+x \right ) y^{\prime }+\left (4+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(x^2-2*x+1)*diff(diff(y(x),x),x)-x*(x+3)*diff(y(x),x)+(x+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+5 x +17 x^{2}+\frac {143}{3} x^{3}+\frac {355}{3} x^{4}+\frac {4043}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {29}{2} x^{2}-\frac {859}{18} x^{3}-\frac {4693}{36} x^{4}-\frac {285181}{900} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 118

ode=x^2*(1-2*x+x^2)*D[y[x],{x,2}] -x*(3+x)*D[y[x],x]+(4+x)*y[x] == 0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {4043 x^5}{15}+\frac {355 x^4}{3}+\frac {143 x^3}{3}+17 x^2+5 x+1\right ) x^2+c_2 \left (\left (-\frac {285181 x^5}{900}-\frac {4693 x^4}{36}-\frac {859 x^3}{18}-\frac {29 x^2}{2}-3 x\right ) x^2+\left (\frac {4043 x^5}{15}+\frac {355 x^4}{3}+\frac {143 x^3}{3}+17 x^2+5 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 1.133 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 - 2*x + 1)*Derivative(y(x), (x, 2)) - x*(x + 3)*Derivative(y(x), x) + (x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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