30.13.4 problem 4

Internal problem ID [7636]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 04:55:13 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }-6 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 60

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

\[ y = \frac {c_1 \left (1+\frac {3}{4} x^{2}-\frac {1}{10} x^{3}+\frac {17}{80} x^{4}-\frac {9}{100} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_2 \left (\ln \left (x \right ) \left (6 x^{2}+\frac {9}{2} x^{4}-\frac {3}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-12 x -24 x^{2}-22 x^{3}-\frac {171}{8} x^{4}-\frac {653}{100} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 73

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

\[ y(x)\to c_2 \left (\frac {7 x^4}{20}-\frac {x^3}{6}+x^2+1\right )+c_1 \left (\frac {1}{3} \left (x^3-6 x^2-6\right ) \log (x)+\frac {7 x^4+240 x^3+72 x^2+180 x+36}{36 x}\right ) \]

✓ Sympy. Time used: 0.342 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} \left (\frac {9 x^{5}}{350} + \frac {3 x^{4}}{20} + \frac {3 x^{3}}{5} + \frac {3 x^{2}}{2} + 2 x + 1\right ) + O\left (x^{6}\right ) \]