83.14.4 problem 2 (c)

Internal problem ID [22019]
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 (c)
Date solved : Thursday, October 02, 2025 at 08:21:43 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {5}{4} x +\frac {35}{32} x^{2}+\frac {105}{128} x^{3}+\frac {1155}{2048} x^{4}+\frac {3003}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-x -2 x^{2}-2 x^{3}-\frac {8}{5} x^{4}-\frac {8}{7} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 79

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

\[ y(x)\to c_2 \left (-\frac {8 x^5}{7}-\frac {8 x^4}{5}-2 x^3-2 x^2-x+1\right )+c_1 \left (\frac {3003 x^5}{8192}+\frac {1155 x^4}{2048}+\frac {105 x^3}{128}+\frac {35 x^2}{32}+\frac {5 x}{4}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.360 (sec). Leaf size: 14

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

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