14.25.18 problem 19

Internal problem ID [4023]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 07:00:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{2}/{3}} \left (1+\frac {2}{5} x -\frac {3}{40} x^{2}-\frac {43}{660} x^{3}+\frac {31}{3696} x^{4}+\frac {2259}{261800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+2 x +\frac {1}{2} x^{2}-\frac {5}{21} x^{3}-\frac {73}{840} x^{4}+\frac {827}{27300} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 83

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

\[ y(x)\to c_2 \left (\frac {827 x^5}{27300}-\frac {73 x^4}{840}-\frac {5 x^3}{21}+\frac {x^2}{2}+2 x+1\right )+c_1 x^{2/3} \left (\frac {2259 x^5}{261800}+\frac {31 x^4}{3696}-\frac {43 x^3}{660}-\frac {3 x^2}{40}+\frac {2 x}{5}+1\right ) \]

✓ Sympy. Time used: 0.475 (sec). Leaf size: 71

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

\[ y{\left (x \right )} = C_{2} \left (\frac {827 x^{5}}{27300} - \frac {73 x^{4}}{840} - \frac {5 x^{3}}{21} + \frac {x^{2}}{2} + 2 x + 1\right ) + C_{1} x^{\frac {2}{3}} \left (\frac {31 x^{4}}{3696} - \frac {43 x^{3}}{660} - \frac {3 x^{2}}{40} + \frac {2 x}{5} + 1\right ) + O\left (x^{6}\right ) \]