7.16.6 problem 6

Internal problem ID [503]
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 : 6
Date solved : Saturday, March 29, 2025 at 04:55:15 PM
CAS classification : [_Laguerre]

\begin{align*} 2 x y^{\prime \prime }-\left (6+2 x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 56

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

\[ y = c_1 \,x^{4} \left (1+\frac {7}{10} x +\frac {21}{80} x^{2}+\frac {11}{160} x^{3}+\frac {143}{10240} x^{4}+\frac {143}{61440} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {15}{16} x^{4}-\frac {21}{32} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-24 x +3 x^{2}-\frac {3}{2} x^{3}-x^{4}-\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 76

ode=2*x*D[y[x],{x,2}]-(6+2*x)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {5}{768} x^4 \log (x)+\frac {49 x^4+96 x^3-192 x^2+1536 x+9216}{9216}\right )+c_2 \left (\frac {143 x^8}{10240}+\frac {11 x^7}{160}+\frac {21 x^6}{80}+\frac {7 x^5}{10}+x^4\right ) \]

✓ Sympy. Time used: 0.791 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} x^{4} \left (\frac {7 x}{10} + 1\right ) + O\left (x^{6}\right ) \]