84.34.8 problem 20.21

Internal problem ID [22335]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 20. Regular singular points and the method of Frobenius. Supplementary problems
Problem number : 20.21
Date solved : Thursday, October 02, 2025 at 08:37:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 62

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

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 103

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

\[ y(x)\to c_2 \left (\frac {x^{9/2}}{384}-\frac {x^{7/2}}{48}+\frac {x^{5/2}}{8}-\frac {x^{3/2}}{2}+\sqrt {x}\right )+c_1 \left (\frac {1}{96} \sqrt {x} \left (x^3-6 x^2+24 x-48\right ) \log (x)-\frac {11 x^4-54 x^3+144 x^2-576}{576 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.345 (sec). Leaf size: 32

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

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