15.23.21 problem 25

Internal problem ID [3371]
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 : 25
Date solved : Sunday, March 30, 2025 at 01:38:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 x y^{\prime }-\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x^3*(x^2+3)*diff(diff(y(x),x),x)+5*x*diff(y(x),x)-(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 99

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

\[ y(x)\to c_1 \left (\frac {18968303719 x^5}{1220703125000}-\frac {20383193 x^4}{1953125000}+\frac {26731 x^3}{3906250}+\frac {259 x^2}{31250}+\frac {37 x}{125}+1\right ) \sqrt [5]{x}+c_2 e^{\left .\frac {5}{3}\right /x} \left (\frac {869909160612721304 x^5}{27030487060546875}+\frac {46847788879262 x^4}{4805419921875}+\frac {15542572604 x^3}{4271484375}+\frac {2270672 x^2}{1265625}+\frac {1372 x}{1125}+1\right ) x^{9/5} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*(x**2 + 3)*Derivative(y(x), (x, 2)) + 5*x*Derivative(y(x), x) - (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**3*(x**2 + 3)*Derivative(y(x), (x, 2)) + 5*x*Derivative(y(x), x) - (x + 1)*y(x) does not match hint 2nd_power_series_regular