12.16.43 problem 39

Internal problem ID [2105]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 39
Date solved : Tuesday, March 04, 2025 at 01:50:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 54

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

\[ y = c_1 \,x^{4} \left (1+\frac {2}{5} x -\frac {8}{5} x^{2}-\frac {86}{105} x^{3}+\frac {445}{168} x^{4}+\frac {9571}{6300} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (24 x^{4}+\frac {48}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+96 x -48 x^{2}+210 x^{4}+\frac {1812}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 71

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

\[ y(x)\to c_1 \left (\frac {1}{12} \left (3 x^4+4 x^2-8 x+12\right )-\frac {1}{6} x^4 \log (x)\right )+c_2 \left (\frac {445 x^8}{168}-\frac {86 x^7}{105}-\frac {8 x^6}{5}+\frac {2 x^5}{5}+x^4\right ) \]

✓ Sympy. Time used: 1.186 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x**2 + 1)*Derivative(y(x), (x, 2)) - x*(x**2 + 3)*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)
 

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