68.17.5 problem 5

Internal problem ID [17823]
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 : 5
Date solved : Thursday, October 02, 2025 at 02:28:34 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x y^{\prime \prime }-5 y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{7}/{2}} \left (1+\frac {1}{3} x +\frac {1}{22} x^{2}+\frac {1}{286} x^{3}+\frac {1}{5720} x^{4}+\frac {3}{486200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {3}{5} x +\frac {3}{10} x^{2}-\frac {3}{10} x^{3}-\frac {9}{40} x^{4}-\frac {9}{200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 85

ode=2*x*D[y[x],{x,2}]-5*D[y[x],x]-3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {9 x^5}{200}-\frac {9 x^4}{40}-\frac {3 x^3}{10}+\frac {3 x^2}{10}-\frac {3 x}{5}+1\right )+c_1 \left (\frac {3 x^5}{486200}+\frac {x^4}{5720}+\frac {x^3}{286}+\frac {x^2}{22}+\frac {x}{3}+1\right ) x^{7/2} \]

✓ Sympy. Time used: 0.241 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {9 x^{5}}{200} - \frac {9 x^{4}}{40} - \frac {3 x^{3}}{10} + \frac {3 x^{2}}{10} - \frac {3 x}{5} + 1\right ) + C_{1} x^{\frac {7}{2}} \left (\frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]