40.17.1 problem 11

Internal problem ID [6802]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 11
Date solved : Sunday, March 30, 2025 at 11:23:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 40

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

\[ y = \left (-x^{5}+x^{4}-x^{3}+x^{2}-x +1\right ) \left (c_1 \sqrt {x}+c_2 x \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 58

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

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

✓ Sympy. Time used: 1.129 (sec). Leaf size: 15

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

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