34.4.14 problem 14

Internal problem ID [6068]
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 : 14
Date solved : Sunday, March 30, 2025 at 10:37:40 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} \left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y&=3 x^{2} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 50

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

\[ y = c_1 \left (1-\frac {2}{3} x +\frac {1}{6} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2+8 x -12 x^{2}+8 x^{3}-2 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}}+x^{3} \left (\frac {1}{5}+\frac {1}{30} x +\frac {1}{105} x^{2}+\operatorname {O}\left (x^{3}\right )\right ) \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 91

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

\[ y(x)\to c_1 \left (\frac {x^2}{6}-\frac {2 x}{3}+1\right )+\frac {c_2 (1-4 x)}{x^2}+\frac {(1-4 x) \left (-\frac {5 x^6}{6}-\frac {3 x^5}{10}\right )}{x^2}+\left (\frac {x^2}{6}-\frac {2 x}{3}+1\right ) \left (-5 x^6-\frac {9 x^5}{5}+\frac {x^3}{2}\right ) \]

✗ Sympy

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

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