83.14.5 problem 2 (d)

Internal problem ID [22020]
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 (d)
Date solved : Thursday, October 02, 2025 at 08:21:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}-3 x \right ) y^{\prime }+\left (4+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(x^2-3*x)*diff(y(x),x)+(x+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-3 x +3 x^{2}-\frac {5}{3} x^{3}+\frac {5}{8} x^{4}-\frac {7}{40} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (5 x -\frac {29}{4} x^{2}+\frac {173}{36} x^{3}-\frac {193}{96} x^{4}+\frac {1459}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) x^{2} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 118

ode=(x^2)*D[y[x],{x,2}]+(x^2-3*x)*D[y[x],x]+(x+4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {7 x^5}{40}+\frac {5 x^4}{8}-\frac {5 x^3}{3}+3 x^2-3 x+1\right ) x^2+c_2 \left (\left (\frac {1459 x^5}{2400}-\frac {193 x^4}{96}+\frac {173 x^3}{36}-\frac {29 x^2}{4}+5 x\right ) x^2+\left (-\frac {7 x^5}{40}+\frac {5 x^4}{8}-\frac {5 x^3}{3}+3 x^2-3 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.313 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x + 4)*y(x) + (x**2 - 3*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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