28.3.11 problem 13

Internal problem ID [7185]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 04:25:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{2} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 66

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

\[ y(x)\to c_1 \left (\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^6}{840}+\frac {x^5}{120}+\frac {x^4}{20}+\frac {x^3}{4}+x^2\right ) \]

✓ Sympy. Time used: 0.318 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {x^{3}}{120} + \frac {x^{2}}{20} + \frac {x}{4} + 1\right ) + \frac {C_{1} \left (\frac {x^{2}}{2} + x + 1\right )}{x} + O\left (x^{6}\right ) \]