7.15.21 problem 21

Internal problem ID [477]
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 : 21
Date solved : Saturday, March 29, 2025 at 04:54:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x y^{\prime }-\left (2 x^{2}+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: 33

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

\[ y = \frac {c_1 \left (1+x^{2}+\frac {1}{10} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1+\frac {1}{7} x^{2}+\frac {1}{154} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 44

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

\[ y(x)\to c_1 x \left (\frac {x^4}{154}+\frac {x^2}{7}+1\right )+\frac {c_2 \left (\frac {x^4}{10}+x^2+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.896 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{154} + \frac {x^{2}}{7} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{10} + x^{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]