15.23.3 problem 3

Internal problem ID [3353]
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 : 3
Date solved : Sunday, March 30, 2025 at 01:37:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (4+x \right ) y^{\prime \prime }+7 x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 40

Order:=6; 
ode:=x^2*(4+x)*diff(diff(y(x),x),x)+7*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{4}} \left (1+\frac {1}{48} x -\frac {5}{19968} x^{2}+\frac {25}{1810432} x^{3}-\frac {75}{62390272} x^{4}+\frac {39}{293601280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+2 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 58

ode=x^2*(4+x)*D[y[x],{x,2}]+7*x*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {39 x^5}{293601280}-\frac {75 x^4}{62390272}+\frac {25 x^3}{1810432}-\frac {5 x^2}{19968}+\frac {x}{48}+1\right )+\frac {c_2 (2 x+1)}{x} \]

✓ Sympy. Time used: 0.970 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 4)*Derivative(y(x), (x, 2)) + 7*x*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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