6.11.4 problem 14

Internal problem ID [1843]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 05:20:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 57

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

\[ y = \left (1-\frac {3}{2} x^{2}+x^{3}-\frac {1}{8} x^{4}-\frac {1}{4} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}-\frac {2}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (-\frac {x^5}{4}-\frac {x^4}{8}+x^3-\frac {3 x^2}{2}+1\right )+c_2 \left (-\frac {2 x^5}{15}+\frac {x^4}{12}+\frac {x^3}{3}-x^2+x\right ) \]

✓ Sympy. Time used: 0.416 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{4}}{8} + x^{3} - \frac {3 x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3}}{12} + \frac {x^{2}}{3} - x + 1\right ) + O\left (x^{6}\right ) \]