49.21.16 problem 6(b)

Internal problem ID [7746]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 190
Problem number : 6(b)
Date solved : Wednesday, March 05, 2025 at 04:54:30 AM
CAS classification : [[_homogeneous, `class C`], _rational, _Riccati]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 21

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

\[ y = 3+\tan \left (\frac {\ln \left (x +2\right )}{2}+\frac {c_{1}}{2}\right ) \left (x +2\right ) \]

✓ Mathematica. Time used: 0.569 (sec). Leaf size: 86

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

\begin{align*} y(x)\to \frac {2^i (x+2)^i x+(2+3 i) 2^i (x+2)^i-2 i c_1 x-(6+4 i) c_1}{i 2^i (x+2)^i-2 c_1} \\ y(x)\to i x+(3+2 i) \\ \end{align*}

✓ Sympy. Time used: 0.518 (sec). Leaf size: 49

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

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