15.24.10 problem 10

Internal problem ID [3382]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 10
Date solved : Sunday, March 30, 2025 at 01:38:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{2} \left (1-x +\frac {9}{10} x^{2}-\frac {17}{30} x^{3}+\frac {251}{840} x^{4}-\frac {37}{280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-12 x -18 x^{2}+44 x^{3}-\frac {115}{2} x^{4}+\frac {477}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (-\frac {115 x^3}{24}+\frac {11 x^2}{3}-\frac {3 x}{2}+\frac {1}{x}-1\right )+c_2 \left (\frac {251 x^6}{840}-\frac {17 x^5}{30}+\frac {9 x^4}{10}-x^3+x^2\right ) \]

✓ Sympy. Time used: 0.997 (sec). Leaf size: 61

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {17 x^{3}}{30} + \frac {9 x^{2}}{10} - x + 1\right ) + \frac {C_{1} \left (- \frac {39 x^{6}}{80} + \frac {27 x^{5}}{40} - \frac {9 x^{4}}{8} - \frac {3 x^{2}}{2} - x + 1\right )}{x} + O\left (x^{6}\right ) \]