6.12.31 problem 37

Internal problem ID [1885]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 37
Date solved : Tuesday, September 30, 2025 at 05:20:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{3}+1\right ) y^{\prime \prime }+15 x^{2} y^{\prime }-36 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

Order:=6; 
ode:=(-x^3+1)*diff(diff(y(x),x),x)+15*x^2*diff(y(x),x)-36*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (6 x^{3}+1\right ) y \left (0\right )+\left (x +\frac {7}{4} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=(1-2*x^3)*D[y[x],{x,2}]-10*x^2*D[y[x],x]-8*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.318 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(15*x**2*Derivative(y(x), x) - 36*x*y(x) + (1 - x**3)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (6 x^{3} + 1\right ) + C_{1} x \left (\frac {7 x^{3}}{4} + 1\right ) + O\left (x^{6}\right ) \]