88.26.8 problem 8

Internal problem ID [24228]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 220
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:01:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+8 x y^{\prime }+\left (2 x -3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 62

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+8*x*diff(y(x),x)+(2*x-3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 101

ode=4*x^2*D[y[x],{x,2}]+(8*x)*D[y[x],{x,1}]+(2*x-3)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^{9/2}}{138240}-\frac {x^{7/2}}{2880}+\frac {x^{5/2}}{96}-\frac {x^{3/2}}{6}+\sqrt {x}\right )+c_1 \left (\frac {31 x^4-352 x^3+576 x^2+4608 x+9216}{9216 x^{3/2}}-\frac {1}{768} \sqrt {x} \left (x^2-16 x+96\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.271 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 8*x*Derivative(y(x), x) + (2*x - 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {x^{4}}{138240} - \frac {x^{3}}{2880} + \frac {x^{2}}{96} - \frac {x}{6} + 1\right ) + O\left (x^{6}\right ) \]