34.3.4 problem 4

Internal problem ID [6043]
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 : 4
Date solved : Sunday, March 30, 2025 at 10:36:58 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y&=2 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 50

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+2*diff(y(x),x)+a^3*x^2*y(x) = 2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{12} a^{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{6} a^{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x}+x \left (1-\frac {1}{20} a^{3} x^{3}+\operatorname {O}\left (x^{5}\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 136

ode=x*D[y[x],{x,2}]+2*D[y[x],x]+a^3*x^2*y[x]==2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {a^6 x^6}{504}-\frac {a^3 x^3}{12}+1\right )+\frac {c_2 \left (\frac {a^6 x^6}{180}-\frac {a^3 x^3}{6}+1\right )}{x}+\left (2 x-\frac {a^3 x^4}{12}\right ) \left (\frac {a^6 x^6}{504}-\frac {a^3 x^3}{12}+1\right )+\frac {\left (\frac {a^3 x^5}{30}-x^2\right ) \left (\frac {a^6 x^6}{180}-\frac {a^3 x^3}{6}+1\right )}{x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**3*x**2*y(x) + x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x) - 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE a**3*x**2*y(x) + x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x) - 2 does not match hint 2nd_power_series_regular