34.4.9 problem 9

Internal problem ID [6063]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 9
Date solved : Sunday, March 30, 2025 at 10:37:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 60

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-(x^2+4*x)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_1 \,x^{3} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\left (6 x^{3}+6 x^{4}+3 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (12-6 x +6 x^{2}+11 x^{3}+5 x^{4}+x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 74

ode=x^2*D[y[x],{x,2}]-(x^2+4*x)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{2} (x+1) x^4 \log (x)+\frac {1}{4} \left (x^4+3 x^3+2 x^2-2 x+4\right ) x\right )+c_2 \left (\frac {x^8}{24}+\frac {x^7}{6}+\frac {x^6}{2}+x^5+x^4\right ) \]

✓ Sympy. Time used: 0.814 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (x**2 + 4*x)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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