40.17.6 problem 16

Internal problem ID [6807]
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 : 16
Date solved : Sunday, March 30, 2025 at 11:23:19 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 56

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

\[ y = c_1 \,x^{3} \left (1-\frac {1}{4} x +\frac {1}{40} x^{2}-\frac {1}{720} x^{3}+\frac {1}{20160} x^{4}-\frac {1}{806400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x^{3}+\frac {1}{4} x^{4}-\frac {1}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +3 x^{2}-\frac {5}{16} x^{4}+\frac {39}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 79

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

\[ y(x)\to c_1 \left (\frac {1}{48} (x-4) x^3 \log (x)+\frac {1}{576} \left (-19 x^4+16 x^3+144 x^2+288 x+576\right )\right )+c_2 \left (\frac {x^7}{20160}-\frac {x^6}{720}+\frac {x^5}{40}-\frac {x^4}{4}+x^3\right ) \]

✓ Sympy. Time used: 0.834 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} x^{3} \left (\frac {x^{2}}{40} - \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]