68.17.3 problem 3

Internal problem ID [17821]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 3
Date solved : Thursday, October 02, 2025 at 02:28:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-3 x -4\right ) y^{\prime \prime }-\left (1+x \right ) y^{\prime }+\left (x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

Order:=6; 
ode:=(x^2-3*x-4)*diff(diff(y(x),x),x)-(1+x)*diff(y(x),x)+(x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{8} x^{2}+\frac {1}{24} x^{3}+\frac {1}{192} x^{4}-\frac {1}{640} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{8} x^{2}-\frac {1}{24} x^{3}+\frac {1}{48} x^{4}+\frac {1}{960} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 70

ode=(x^2-3*x-4)*D[y[x],{x,2}]-(x+1)*D[y[x],x]+(x^2-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{640}+\frac {x^4}{192}+\frac {x^3}{24}-\frac {x^2}{8}+1\right )+c_2 \left (\frac {x^5}{960}+\frac {x^4}{48}-\frac {x^3}{24}-\frac {x^2}{8}+x\right ) \]

✓ Sympy. Time used: 0.383 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x - 1)*Derivative(y(x), x) + (x**2 - 1)*y(x) + (x**2 - 3*x - 4)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = - \frac {7 x^{4} r{\left (3 \right )}}{16} + \frac {39 x^{5} r{\left (3 \right )}}{160} + C_{2} \left (- \frac {3 x^{5}}{256} + \frac {3 x^{4}}{128} - \frac {x^{2}}{8} + 1\right ) + C_{1} x \left (\frac {43 x^{4}}{3840} + \frac {x^{3}}{384} - \frac {x}{8} + 1\right ) + O\left (x^{6}\right ) \]