85.75.8 problem 1 (h)

Internal problem ID [22973]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 329
Problem number : 1 (h)
Date solved : Thursday, October 02, 2025 at 09:17:06 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 28

Order:=6; 
ode:=x*diff(diff(y(x),x),x)-diff(y(x),x)+4*x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {4}{15} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-2+\frac {8}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 30

ode=x*D[y[x],{x,2}]-D[y[x],x]+4*x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {4 x^3}{3}\right )+c_2 \left (x^2-\frac {4 x^5}{15}\right ) \]

✓ Sympy. Time used: 0.266 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*y(x) + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {4 x^{3}}{3}\right ) + C_{1} x^{2} \left (1 - \frac {4 x^{3}}{15}\right ) + O\left (x^{6}\right ) \]