20.26.33 problem 27

Internal problem ID [4058]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 27
Date solved : Sunday, March 30, 2025 at 02:15:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 62

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

\[ y = \frac {c_1 \,x^{2} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 101

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

\[ y(x)\to c_2 \left (\frac {x^{11/2}}{8640}+\frac {x^{9/2}}{360}+\frac {x^{7/2}}{24}+\frac {x^{5/2}}{3}+x^{3/2}\right )+c_1 \left (\frac {31 x^4+176 x^3+144 x^2-576 x+576}{576 \sqrt {x}}-\frac {1}{48} x^{3/2} \left (x^2+8 x+24\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.812 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} x^{\frac {3}{2}} \left (\frac {x^{3}}{360} + \frac {x^{2}}{24} + \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]