6.16.5 problem 1

Internal problem ID [2067]
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 : 1
Date solved : Tuesday, September 30, 2025 at 05:23:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 60

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

\[ y = \left (c_1 \,x^{2} \left (1-\frac {4}{3} x +\frac {2}{3} x^{2}-\frac {8}{45} x^{3}+\frac {4}{135} x^{4}-\frac {16}{4725} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\left (16 x^{2}-\frac {64}{3} x^{3}+\frac {32}{3} x^{4}-\frac {128}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (-2-8 x +\frac {256}{9} x^{3}-\frac {200}{9} x^{4}+\frac {5024}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {1}{9} x \left (124 x^4-176 x^3+36 x^2+36 x+9\right )-\frac {8}{3} x^3 \left (2 x^2-4 x+3\right ) \log (x)\right )+c_2 \left (\frac {4 x^7}{135}-\frac {8 x^6}{45}+\frac {2 x^5}{3}-\frac {4 x^4}{3}+x^3\right ) \]

✓ Sympy. Time used: 0.255 (sec). Leaf size: 24

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

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