83.14.3 problem 2 (b)

Internal problem ID [22018]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter X. Solution in power series. Ex. XVIII at page 174
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 08:21:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }+3 x y^{\prime }-6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

Order:=6; 
ode:=(2*x^2+1)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)-6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (-x^{4}+3 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}-\frac {3}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

ode=(2*x^2+1)*D[y[x],{x,2}]+3*x*D[y[x],x]-6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {3 x^5}{8}+\frac {x^3}{2}+x\right )+c_1 \left (-x^4+3 x^2+1\right ) \]

✓ Sympy. Time used: 0.277 (sec). Leaf size: 27

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

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