7.15.32 problem 32

Internal problem ID [488]
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 : 32
Date solved : Saturday, March 29, 2025 at 04:54:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 37

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

\[ y = \frac {c_1 \left (1-\frac {5}{2} x -\frac {15}{8} x^{2}-\frac {5}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{8960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1+\frac {1}{5} x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 58

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

\[ y(x)\to \frac {c_2 \left (-\frac {x^5}{8960}+\frac {x^4}{384}-\frac {5 x^3}{48}-\frac {15 x^2}{8}-\frac {5 x}{2}+1\right )}{\sqrt {x}}+c_1 \left (\frac {x}{5}+1\right ) x \]

✓ Sympy. Time used: 0.944 (sec). Leaf size: 51

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x}{5} + 1\right ) + \frac {C_{1} \left (- \frac {x^{5}}{8960} + \frac {x^{4}}{384} - \frac {5 x^{3}}{48} - \frac {15 x^{2}}{8} - \frac {5 x}{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]