6.14.8 problem 5

Internal problem ID [1949]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:21:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 12 x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 47

Order:=6; 
ode:=12*x^2*(1+x)*diff(diff(y(x),x),x)+x*(3*x^2+35*x+11)*diff(y(x),x)-(-5*x^2-10*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-x +\frac {7}{8} x^{2}-\frac {19}{24} x^{3}+\frac {283}{384} x^{4}-\frac {1339}{1920} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{4}}}+c_2 \,x^{{1}/{3}} \left (1-x +\frac {28}{31} x^{2}-\frac {1111}{1333} x^{3}+\frac {57493}{73315} x^{4}-\frac {3668716}{4912105} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 86

ode=12*x^2*(1+x)*D[y[x],{x,2}]+x*(11+35*x+3*x^2)*D[y[x],x]-(1-10*x-5*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {3668716 x^5}{4912105}+\frac {57493 x^4}{73315}-\frac {1111 x^3}{1333}+\frac {28 x^2}{31}-x+1\right )+\frac {c_2 \left (-\frac {1339 x^5}{1920}+\frac {283 x^4}{384}-\frac {19 x^3}{24}+\frac {7 x^2}{8}-x+1\right )}{\sqrt [4]{x}} \]

✓ Sympy. Time used: 0.576 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(3*x**2 + 35*x + 11)*Derivative(y(x), x) - (-5*x**2 - 10*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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