6.14.16 problem 16

Internal problem ID [1957]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 05:21:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 48

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

\[ y = \frac {c_2 \,x^{{5}/{2}} \left (1-\frac {1}{2} x +\frac {1}{8} x^{2}-\frac {1}{48} x^{3}+\frac {1}{384} x^{4}-\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {1}{3} x +\frac {1}{3} x^{2}-\frac {1}{3} x^{3}+\frac {1}{9} x^{4}-\frac {1}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 88

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{3840}+\frac {x^4}{384}-\frac {x^3}{48}+\frac {x^2}{8}-\frac {x}{2}+1\right )+\frac {c_2 \left (-\frac {x^5}{45}+\frac {x^4}{9}-\frac {x^3}{3}+\frac {x^2}{3}+\frac {x}{3}+1\right )}{x^2} \]

✓ Sympy. Time used: 0.395 (sec). Leaf size: 73

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

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {x^{4}}{384} - \frac {x^{3}}{48} + \frac {x^{2}}{8} - \frac {x}{2} + 1\right ) + \frac {C_{1} \left (- \frac {x^{7}}{2835} + \frac {x^{6}}{315} - \frac {x^{5}}{45} + \frac {x^{4}}{9} - \frac {x^{3}}{3} + \frac {x^{2}}{3} + \frac {x}{3} + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]