40.17.5 problem 15

Internal problem ID [6806]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 15
Date solved : Sunday, March 30, 2025 at 11:23:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 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: 0.020 (sec). Leaf size: 34

Order:=6; 
ode:=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);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 65

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

\[ y(x)\to c_1 x \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (x \left (\frac {x^2}{4}-\frac {3 x^4}{128}\right )+x \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.945 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(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)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{64} - \frac {x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]