15.23.19 problem 23

Internal problem ID [3369]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 23
Date solved : Sunday, March 30, 2025 at 01:38:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 33

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1-\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 48

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

\[ y(x)\to c_1 x \left (\frac {x^4}{360}-\frac {x^2}{10}+1\right )+c_2 \sqrt {x} \left (\frac {x^4}{168}-\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.921 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{360} - \frac {x^{2}}{10} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{168} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]