15.24.5 problem 5

Internal problem ID [3377]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 5
Date solved : Tuesday, March 04, 2025 at 04:37:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 48

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

\[ y \left (x \right ) = x^{2} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+4 x +6 x^{2}+\frac {16}{3} x^{3}+\frac {10}{3} x^{4}+\frac {8}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-6\right ) x -13 x^{2}-\frac {124}{9} x^{3}-\frac {173}{18} x^{4}-\frac {374}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 116

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

\[ y(x)\to c_1 \left (\frac {8 x^5}{5}+\frac {10 x^4}{3}+\frac {16 x^3}{3}+6 x^2+4 x+1\right ) x^2+c_2 \left (\left (-\frac {374 x^5}{75}-\frac {173 x^4}{18}-\frac {124 x^3}{9}-13 x^2-6 x\right ) x^2+\left (\frac {8 x^5}{5}+\frac {10 x^4}{3}+\frac {16 x^3}{3}+6 x^2+4 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.780 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {16 x^{3}}{3} + 6 x^{2} + 4 x + 1\right ) + O\left (x^{6}\right ) \]