12.16.9 problem 5

Internal problem ID [2071]
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 : 5
Date solved : Tuesday, March 04, 2025 at 01:49:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2+3 x \right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 64

Order:=6; 
ode:=2*x^2*(3*x+2)*diff(diff(y(x),x),x)+x*(4+21*x)*diff(y(x),x)-(1-9*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 x \left (1-\frac {9}{4} x +\frac {135}{32} x^{2}-\frac {945}{128} x^{3}+\frac {25515}{2048} x^{4}-\frac {168399}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {3}{4} x +\frac {27}{16} x^{2}-\frac {405}{128} x^{3}+\frac {2835}{512} x^{4}-\frac {76545}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {15}{4} x +\frac {63}{8} x^{2}-\frac {3699}{256} x^{3}+\frac {25623}{1024} x^{4}-\frac {1375137}{32768} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 108

ode=2*x^2*(2+3*x)*D[y[x],{x,2}]+x*(4+21*x)*D[y[x],x]-(1-9*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {25515 x^{9/2}}{2048}-\frac {945 x^{7/2}}{128}+\frac {135 x^{5/2}}{32}-\frac {9 x^{3/2}}{4}+\sqrt {x}\right )+c_1 \left (\frac {3}{512} \sqrt {x} \left (945 x^3-540 x^2+288 x-128\right ) \log (x)+\frac {8613 x^4-5076 x^3+2880 x^2-1536 x+1024}{1024 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 1.041 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(3*x + 2)*Derivative(y(x), (x, 2)) + x*(21*x + 4)*Derivative(y(x), x) - (1 - 9*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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