78.18.7 problem 7

Internal problem ID [18348]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 28. Second Order Linear Equations. Ordinary Points. Problems at page 217
Problem number : 7
Date solved : Monday, March 31, 2025 at 05:26:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 55

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

\[ y = \left (1-n \,x^{2}+\frac {n \left (n -2\right ) x^{4}}{6}\right ) y \left (0\right )+\left (x -\frac {\left (n -1\right ) x^{3}}{3}+\frac {\left (n^{2}-4 n +3\right ) x^{5}}{30}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 78

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

\[ y(x)\to c_2 \left (\frac {n^2 x^5}{30}-\frac {2 n x^5}{15}-\frac {n x^3}{3}+\frac {x^5}{10}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {n^2 x^4}{6}-\frac {n x^4}{3}-n x^2+1\right ) \]

✓ Sympy. Time used: 0.736 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \left (\frac {n^{2} x^{4}}{6} - \frac {n x^{4}}{3} - n x^{2} + 1\right ) + C_{1} x \left (- \frac {n x^{2}}{3} + \frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]