6.15.41 problem 37

Internal problem ID [2039]
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 : 37
Date solved : Tuesday, September 30, 2025 at 05:22:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 36

Order:=6; 
ode:=9*x^2*diff(diff(y(x),x),x)-3*x*(-2*x^2+7)*diff(y(x),x)+(2*x^2+25)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{{5}/{3}} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{3} x^{2}+\frac {1}{18} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{6} x^{2}-\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 77

ode=9*x^2*D[y[x],{x,2}]-3*x*(7-2*x^2)*D[y[x],x]+(25+2*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^4}{18}-\frac {x^2}{3}+1\right ) x^{5/3}+c_2 \left (\left (\frac {x^2}{6}-\frac {x^4}{24}\right ) x^{5/3}+\left (\frac {x^4}{18}-\frac {x^2}{3}+1\right ) x^{5/3} \log (x)\right ) \]

✓ Sympy. Time used: 0.366 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) - 3*x*(7 - 2*x**2)*Derivative(y(x), x) + (2*x**2 + 25)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{\frac {5}{3}} \left (1 - \frac {x^{2}}{3}\right ) + O\left (x^{6}\right ) \]