6.15.22 problem 18

Internal problem ID [2020]
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 II. Exercises 7.6. Page 374
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 05:22:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (9+4 x \right ) y^{\prime \prime }+3 x y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(9+4*x)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{{1}/{3}} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{81} x +\frac {25}{26244} x^{2}-\frac {3025}{19131876} x^{3}+\frac {874225}{24794911296} x^{4}-\frac {18498601}{2008387814976} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {14}{81} x -\frac {35}{2916} x^{2}+\frac {110495}{57395628} x^{3}-\frac {62786185}{148769467776} x^{4}+\frac {1315043653}{12050326889856} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 134

ode=x^2*(9+4*x)*D[y[x],{x,2}]+3*x*D[y[x],x]+(1+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {18498601 x^5}{2008387814976}+\frac {874225 x^4}{24794911296}-\frac {3025 x^3}{19131876}+\frac {25 x^2}{26244}-\frac {x}{81}+1\right )+c_2 \left (\sqrt [3]{x} \left (\frac {1315043653 x^5}{12050326889856}-\frac {62786185 x^4}{148769467776}+\frac {110495 x^3}{57395628}-\frac {35 x^2}{2916}+\frac {14 x}{81}\right )+\sqrt [3]{x} \left (-\frac {18498601 x^5}{2008387814976}+\frac {874225 x^4}{24794911296}-\frac {3025 x^3}{19131876}+\frac {25 x^2}{26244}-\frac {x}{81}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.372 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(4*x + 9)*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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