74.17.10 problem 10

Internal problem ID [16462]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 10
Date solved : Thursday, March 13, 2025 at 08:14:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+2 x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.073 (sec). Leaf size: 60

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

\[ y = c_{1} x \left (1-\frac {3}{2} x +\frac {5}{4} x^{2}-\frac {35}{48} x^{3}+\frac {21}{64} x^{4}-\frac {77}{640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {3}{2} x^{2}-\frac {5}{4} x^{3}+\frac {35}{48} x^{4}-\frac {21}{64} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-2 x +\frac {7}{4} x^{2}-\frac {11}{12} x^{3}+\frac {61}{192} x^{4}-\frac {131}{1920} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {1}{48} x \left (35 x^3-60 x^2+72 x-48\right ) \log (x)+\frac {1}{192} \left (-79 x^4+64 x^3+48 x^2-192 x+192\right )\right )+c_2 \left (\frac {21 x^5}{64}-\frac {35 x^4}{48}+\frac {5 x^3}{4}-\frac {3 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.748 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + x*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)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {21 x^{4}}{64} - \frac {35 x^{3}}{48} + \frac {5 x^{2}}{4} - \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]