56.4.38 problem 35

Internal problem ID [8927]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 35
Date solved : Sunday, March 30, 2025 at 01:55:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {3}{2} x^{2}-\frac {2}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (3 x -\frac {13}{4} x^{2}+\frac {31}{18} x^{3}-\frac {173}{288} x^{4}+\frac {187}{1200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \left (-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right )+c_2 \left (\frac {187 x^5}{1200}-\frac {173 x^4}{288}+\frac {31 x^3}{18}-\frac {13 x^2}{4}+\left (-\frac {x^5}{20}+\frac {5 x^4}{24}-\frac {2 x^3}{3}+\frac {3 x^2}{2}-2 x+1\right ) \log (x)+3 x\right ) \]

✓ Sympy. Time used: 0.823 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{5}}{20} + \frac {5 x^{4}}{24} - \frac {2 x^{3}}{3} + \frac {3 x^{2}}{2} - 2 x + 1\right ) + O\left (x^{6}\right ) \]