6.15.27 problem 23

Internal problem ID [2025]
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 : 23
Date solved : Tuesday, September 30, 2025 at 05:22:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 56

Order:=8; 
ode:=x^2*(2*x+1)*diff(diff(y(x),x),x)+x*(5+9*x)*diff(y(x),x)+(4+3*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+3 x +\frac {3}{2} x^{2}-\frac {1}{2} x^{3}+\frac {3}{8} x^{4}-\frac {3}{8} x^{5}+\frac {7}{16} x^{6}-\frac {9}{16} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-5\right ) x -\frac {25}{4} x^{2}+\frac {5}{4} x^{3}-\frac {25}{32} x^{4}+\frac {113}{160} x^{5}-\frac {247}{320} x^{6}+\frac {2123}{2240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x^{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 164

ode=x^2*(1+2*x)*D[y[x],{x,2}]+x*(5+9*x)*D[y[x],x]+(4+3*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {9 x^7}{16}+\frac {7 x^6}{16}-\frac {3 x^5}{8}+\frac {3 x^4}{8}-\frac {x^3}{2}+\frac {3 x^2}{2}+3 x+1\right )}{x^2}+c_2 \left (\frac {\frac {2123 x^7}{2240}-\frac {247 x^6}{320}+\frac {113 x^5}{160}-\frac {25 x^4}{32}+\frac {5 x^3}{4}-\frac {25 x^2}{4}-5 x}{x^2}+\frac {\left (-\frac {9 x^7}{16}+\frac {7 x^6}{16}-\frac {3 x^5}{8}+\frac {3 x^4}{8}-\frac {x^3}{2}+\frac {3 x^2}{2}+3 x+1\right ) \log (x)}{x^2}\right ) \]

✓ Sympy. Time used: 0.425 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) + x*(9*x + 5)*Derivative(y(x), x) + (3*x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x^{2}} + O\left (x^{8}\right ) \]