48.7.4 problem 4

Internal problem ID [9938]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 06:44:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 74

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

\[ y = \frac {c_1 x \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {1}{16} x^{3}+\frac {1}{96} x^{4}+\frac {1}{768} x^{5}+\frac {1}{7680} x^{6}+\frac {1}{92160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1+\frac {1}{2} x -\frac {1}{32} x^{3}-\frac {5}{576} x^{4}-\frac {13}{9216} x^{5}-\frac {77}{460800} x^{6}-\frac {29}{1843200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 141

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

\[ y(x)\to c_2 \left (\frac {x^{13/2}}{46080}+\frac {x^{11/2}}{3840}+\frac {x^{9/2}}{384}+\frac {x^{7/2}}{48}+\frac {x^{5/2}}{8}+\frac {x^{3/2}}{2}+\sqrt {x}\right )+c_1 \left (\frac {\sqrt {x} \left (x^5+10 x^4+80 x^3+480 x^2+1920 x+3840\right ) \log (x)}{7680}-\frac {137 x^6+1250 x^5+8800 x^4+43200 x^3+115200 x^2-460800}{460800 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.331 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {x^{6}}{46080} + \frac {x^{5}}{3840} + \frac {x^{4}}{384} + \frac {x^{3}}{48} + \frac {x^{2}}{8} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]