35.1.9 problem Ex. 8(ii), page 258

Internal problem ID [8108]
Book : A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd. New York, reprinted 1956
Section : Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius. page 243
Problem number : Ex. 8(ii), page 258
Date solved : Tuesday, September 30, 2025 at 05:15:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (x^{3}+1\right ) y^{\prime }+b x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 54

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+(x^3+1)*diff(y(x),x)+b*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} b \,x^{2}+\frac {1}{64} b^{2} x^{4}+\frac {1}{50} b \,x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {b}{4} x^{2}-\frac {1}{9} x^{3}-\frac {3}{128} b^{2} x^{4}-\frac {61}{4500} b \,x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 103

ode=x*D[y[x],{x,2}]+(1+x*x^2)*D[y[x],x]+b*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {b^2 x^4}{64}+\frac {b x^5}{50}-\frac {b x^2}{4}+1\right )+c_2 \left (-\frac {3 b^2 x^4}{128}+\left (\frac {b^2 x^4}{64}+\frac {b x^5}{50}-\frac {b x^2}{4}+1\right ) \log (x)-\frac {61 b x^5}{4500}+\frac {b x^2}{4}-\frac {x^3}{9}\right ) \]

✓ Sympy. Time used: 0.322 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*x*y(x) + x*Derivative(y(x), (x, 2)) + (x**3 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {b^{2} x^{4}}{64} + \frac {b x^{5}}{50} - \frac {b x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]