15.23.17 problem 21

Internal problem ID [3367]
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 : 21
Date solved : Sunday, March 30, 2025 at 01:38:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (8-x \right ) x^{2} y^{\prime \prime }+6 x y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 48

Order:=6; 
ode:=(8-x)*x^2*diff(diff(y(x),x),x)+6*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{3}/{4}} \left (1-\frac {1}{56} x -\frac {3}{9856} x^{2}-\frac {1}{78848} x^{3}-\frac {5}{6848512} x^{4}-\frac {63}{1260126208} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {5}{32} x -\frac {3}{2048} x^{2}-\frac {7}{196608} x^{3}-\frac {539}{327155712} x^{4}-\frac {5929}{59324235776} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{4}}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 90

ode=(8-x)*x^2*D[y[x],{x,2}]+6*x*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {63 x^5}{1260126208}-\frac {5 x^4}{6848512}-\frac {x^3}{78848}-\frac {3 x^2}{9856}-\frac {x}{56}+1\right )+\frac {c_2 \left (-\frac {5929 x^5}{59324235776}-\frac {539 x^4}{327155712}-\frac {7 x^3}{196608}-\frac {3 x^2}{2048}+\frac {5 x}{32}+1\right )}{\sqrt [4]{x}} \]

✓ Sympy. Time used: 0.953 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(8 - x)*Derivative(y(x), (x, 2)) + 6*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)
 

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