6.16.14 problem 10

Internal problem ID [2076]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 05:23:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(1+x)*diff(diff(y(x),x),x)-x*(3+10*x)*diff(y(x),x)+30*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{4} \left (1-\frac {2}{5} x +\operatorname {O}\left (x^{6}\right )\right )+\left (43200 x^{4}-17280 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (-144-1440 x -7200 x^{2}-28800 x^{3}-90720 x^{4}+82944 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 48

ode=x^2*(1+x)*D[y[x],{x,2}]-x*(3+10*x)*D[y[x],x]+30*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x^4-\frac {2 x^5}{5}\right )+c_1 \left (745 x^4-300 x^4 \log (x)+200 x^3+50 x^2+10 x+1\right ) \]

✓ Sympy. Time used: 0.426 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) - x*(10*x + 3)*Derivative(y(x), x) + 30*x*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{4} \left (1 - 6 x\right ) + O\left (x^{6}\right ) \]