15.24.13 problem 13

Internal problem ID [3385]
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 : 13
Date solved : Sunday, March 30, 2025 at 01:38:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (x -7\right ) y^{\prime }+\left (x +12\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 60

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(x-7)*diff(y(x),x)+(x+12)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{2} \left (c_1 \,x^{4} \left (1-\frac {7}{5} x +\frac {14}{15} x^{2}-\frac {2}{5} x^{3}+\frac {1}{8} x^{4}-\frac {11}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\left (360 x^{4}-504 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right )+\left (-144-144 x -144 x^{2}-240 x^{3}+342 x^{4}+54 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 79

ode=x^2*D[y[x],{x,2}]+x*(x-7)*D[y[x],x]+(x+12)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {5}{2} x^6 \log (x)-\frac {1}{12} \left (21 x^4-20 x^3-12 x^2-12 x-12\right ) x^2\right )+c_2 \left (\frac {x^{10}}{8}-\frac {2 x^9}{5}+\frac {14 x^8}{15}-\frac {7 x^7}{5}+x^6\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(x - 7)*Derivative(y(x), x) + (x + 12)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None