15.23.2 problem 2

Internal problem ID [3352]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 04:36:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 44

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

\[ y \left (x \right ) = c_{1} x^{{5}/{3}} \left (1-\frac {7}{8} x +\frac {7}{8} x^{2}-\frac {23}{24} x^{3}+\frac {1817}{1632} x^{4}-\frac {219857}{163200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x -x^{2}+\frac {11}{12} x^{3}-\frac {319}{336} x^{4}+\frac {319}{300} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 79

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

\[ y(x)\to c_2 \left (\frac {319 x^5}{300}-\frac {319 x^4}{336}+\frac {11 x^3}{12}-x^2+x+1\right )+c_1 \left (-\frac {219857 x^5}{163200}+\frac {1817 x^4}{1632}-\frac {23 x^3}{24}+\frac {7 x^2}{8}-\frac {7 x}{8}+1\right ) x^{5/3} \]

✓ Sympy. Time used: 0.994 (sec). Leaf size: 63

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

\[ y{\left (x \right )} = C_{2} \left (\frac {648 x^{5}}{175} - \frac {108 x^{4}}{7} + 36 x^{3} - 36 x^{2} + 6 x + 1\right ) + C_{1} x^{\frac {5}{3}} \left (- \frac {18 x^{3}}{77} + \frac {9 x^{2}}{11} - \frac {3 x}{2} + 1\right ) + O\left (x^{6}\right ) \]