7.16.14 problem 14

Internal problem ID [511]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 14
Date solved : Tuesday, March 04, 2025 at 11:25:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 45

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

\[ y = c_1 \,x^{2} \left (1-\frac {2}{5} x +\frac {1}{10} x^{2}-\frac {2}{105} x^{3}+\frac {1}{336} x^{4}-\frac {1}{2520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+96 x -24 x^{2}+2 x^{4}-\frac {4}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^2}{72}+\frac {1}{x^2}-\frac {2}{3 x}+\frac {1}{6}\right )+c_2 \left (\frac {x^6}{336}-\frac {2 x^5}{105}+\frac {x^4}{10}-\frac {2 x^3}{5}+x^2\right ) \]

✓ Sympy. Time used: 0.877 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {2 x^{3}}{105} + \frac {x^{2}}{10} - \frac {2 x}{5} + 1\right ) + \frac {C_{1} \left (\frac {x^{2}}{6} - \frac {2 x}{3} + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]