67.25.24 problem 35.4 (j)

Internal problem ID [17019]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (j)
Date solved : Thursday, October 02, 2025 at 01:41:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-3 x^{3}+3 x^{2}\right ) y^{\prime \prime }-\left (5 x^{2}+4 x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 47

Order:=6; 
ode:=(-3*x^3+3*x^2)*diff(diff(y(x),x),x)-(5*x^2+4*x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {1}{2} x -2 x^{2}-\frac {7}{2} x^{3}-5 x^{4}-\frac {13}{2} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{2} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 74

ode=(3*x^2-3*x^3)*D[y[x],{x,2}]-(4*x+5*x^2)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right ) x^2+c_2 \left (-\frac {13 x^5}{2}-5 x^4-\frac {7 x^3}{2}-2 x^2-\frac {x}{2}+1\right ) \sqrt [3]{x} \]

✓ Sympy. Time used: 0.397 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-5*x**2 - 4*x)*Derivative(y(x), x) + (-3*x**3 + 3*x**2)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{2} + C_{1} \sqrt [3]{x} + O\left (x^{6}\right ) \]