64.5.7 problem 7

Internal problem ID [13249]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number : 7
Date solved : Monday, March 31, 2025 at 07:43:28 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 24

ode:=x*diff(y(x),x)+(2*x+1)/(1+x)*y(x) = x-1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{3}+3 c_1 -3 x}{3 x \left (x +1\right )} \]

✓ Mathematica. Time used: 0.142 (sec). Leaf size: 80

ode=x*D[y[x],x]+(2*x+1)/(x+1)*y[x]==x-1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {2 K[1]+1}{K[1]^2+K[1]}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {2 K[1]+1}{K[1]^2+K[1]}dK[1]\right ) (K[2]-1)}{K[2]}dK[2]+c_1\right ) \]

✓ Sympy. Time used: 0.320 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - x + 1 + (2*x + 1)*y(x)/(x + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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