68.17.6 problem 6

Internal problem ID [17824]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 6
Date solved : Thursday, October 02, 2025 at 02:28:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 5 x y^{\prime \prime }+8 y^{\prime }-y x&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 32

Order:=6; 
ode:=5*x*diff(diff(y(x),x),x)+8*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {1}{14} x^{2}+\frac {1}{952} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{5}}}+c_2 \left (1+\frac {1}{26} x^{2}+\frac {1}{2392} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \left (\frac {x^4}{2392}+\frac {x^2}{26}+1\right )+\frac {c_2 \left (\frac {x^4}{952}+\frac {x^2}{14}+1\right )}{x^{3/5}} \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + 5*x*Derivative(y(x), (x, 2)) + 8*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{2392} + \frac {x^{2}}{26} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{952} + \frac {x^{2}}{14} + 1\right )}{x^{\frac {3}{5}}} + O\left (x^{6}\right ) \]