15.23.7 problem 7

Internal problem ID [3357]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 04:36:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-x y^{\prime }+\left (1-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 86

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

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

✓ Sympy. Time used: 1.046 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-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)
 

\[ y{\left (x \right )} = C_{2} x + C_{1} \sqrt {x} + O\left (x^{6}\right ) \]