7.15.5 problem 5

Internal problem ID [461]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 5
Date solved : Saturday, March 29, 2025 at 04:54:10 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 60

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

\[ y = \frac {c_1 \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{20} x^{4}-\frac {17}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +\left (\left (-2\right ) x +x^{3}-\frac {1}{6} x^{4}-\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1+2 x -\frac {5}{2} x^{2}-\frac {11}{6} x^{3}+\frac {7}{9} x^{4}+\frac {77}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 77

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

\[ y(x)\to c_2 \left (\frac {x^4}{20}+\frac {x^3}{12}-\frac {x^2}{2}+1\right )+c_1 \left (\frac {37 x^4-120 x^3-90 x^2+180 x+36}{36 x}-\frac {1}{6} \left (x^3-6 x^2+12\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.877 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {9 x^{5}}{3200} + \frac {9 x^{4}}{320} - \frac {3 x^{3}}{16} + \frac {3 x^{2}}{4} - \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]