15.25.15 problem 14

Internal problem ID [3402]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 14
Date solved : Sunday, March 30, 2025 at 01:39:16 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-x y^{\prime }+\left (1-x \right ) y&=x^{2} \left (1+x \right )^{2} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 61

Order:=6; 
ode:=(x^3+2*x^2)*diff(diff(y(x),x),x)-x*diff(y(x),x)+(1-x)*y(x) = x^2*(1+x)^2; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {5}{96} x^{2}-\frac {11}{1152} x^{3}+\frac {341}{129024} x^{4}-\frac {20119}{23224320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1+\frac {1}{3} x -\frac {1}{30} x^{2}+\frac {1}{126} x^{3}-\frac {11}{4536} x^{4}+\frac {19}{22680} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{6} x +\frac {1}{126} x^{2}-\frac {11}{4536} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

✓ Mathematica. Time used: 0.279 (sec). Leaf size: 247

ode=(2*x^2+x^3)*D[y[x],{x,2}]-x*D[y[x],x]+(1-x)*y[x]==x^2*(1+x)^2; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {20119 x^5}{23224320}+\frac {341 x^4}{129024}-\frac {11 x^3}{1152}+\frac {5 x^2}{96}+\frac {5 x}{4}+1\right )+c_2 x \left (\frac {19 x^5}{22680}-\frac {11 x^4}{4536}+\frac {x^3}{126}-\frac {x^2}{30}+\frac {x}{3}+1\right )+\sqrt {x} \left (-\frac {20119 x^5}{23224320}+\frac {341 x^4}{129024}-\frac {11 x^3}{1152}+\frac {5 x^2}{96}+\frac {5 x}{4}+1\right ) \left (\frac {4997 x^{11/2}}{2903040}-\frac {1853 x^{9/2}}{181440}-\frac {183 x^{7/2}}{560}-\frac {5 x^{5/2}}{6}-\frac {2 x^{3/2}}{3}\right )+x \left (\frac {19 x^5}{22680}-\frac {11 x^4}{4536}+\frac {x^3}{126}-\frac {x^2}{30}+\frac {x}{3}+1\right ) \left (\frac {479 x^6}{136080}-\frac {13 x^5}{840}+\frac {13 x^4}{72}+\frac {17 x^3}{18}+\frac {3 x^2}{2}+x\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*(x + 1)**2 - x*Derivative(y(x), x) + (1 - x)*y(x) + (x**3 + 2*x**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 -x**2*(x + 1)**2 - x*Derivative(y(x), x) + (1 - x)*y(x) + (x**3 + 2*x**2)*Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular