6.16.1 problem Example 7.7.1 page 381

Internal problem ID [2063]
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 : Example 7.7.1 page 381
Date solved : Tuesday, September 30, 2025 at 05:23:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 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: 62

Order:=6; 
ode:=2*x^2*(x+2)*diff(diff(y(x),x),x)-x*(4-7*x)*diff(y(x),x)-(5-3*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {7}{4} x +\frac {63}{32} x^{2}-\frac {231}{128} x^{3}+\frac {3003}{2048} x^{4}-\frac {9009}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {45}{32} x^{3}+\frac {315}{128} x^{4}-\frac {2835}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {9}{8} x^{2}-\frac {981}{64} x^{3}+\frac {6417}{256} x^{4}-\frac {28089}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 98

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

\[ y(x)\to c_2 \left (\frac {3003 x^{13/2}}{2048}-\frac {231 x^{11/2}}{128}+\frac {63 x^{9/2}}{32}-\frac {7 x^{7/2}}{4}+x^{5/2}\right )+c_1 \left (\frac {15}{512} (7 x-4) x^{5/2} \log (x)+\frac {809 x^4-548 x^3+96 x^2+128 x+1024}{1024 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.465 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x + 2)*Derivative(y(x), (x, 2)) - x*(4 - 7*x)*Derivative(y(x), x) - (5 - 3*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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