26.6.3 problem Exercise 12.3, page 103

Internal problem ID [7003]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.3, page 103
Date solved : Tuesday, September 30, 2025 at 04:13:30 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 81

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

\[ \frac {\left (-y c_1 +1+c_1 \,x^{2}+\left (2 c_1 +1\right ) x \right ) \sqrt {y+1}-\left (-y c_1 -1+c_1 \,x^{2}+\left (2 c_1 -1\right ) x \right ) \left (x +1\right )}{\left (x^{2}+2 x -y\right ) \left (-\sqrt {y+1}+1+x \right )} = 0 \]

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 214

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\left (\frac {2 (K[1]+1)}{\left (K[1]^2+2 K[1]-K[2]\right )^2}+\frac {2 \sqrt {K[2]+1}}{\left (K[1]^2+2 K[1]-K[2]\right )^2}+\frac {1}{\left (K[1]^2+2 K[1]-K[2]\right ) \sqrt {K[2]+1}}\right )dK[1]+\frac {1}{(-x-1) \sqrt {K[2]+1}}+\frac {\sqrt {K[2]+1}}{(x+1) \left (-x^2-2 x+K[2]\right )}+\frac {1}{-x^2-2 x+K[2]}\right )dK[2]+\int _1^x\left (\frac {2 (K[1]+1)}{K[1]^2+2 K[1]-y(x)}+\frac {2 \sqrt {y(x)+1}}{K[1]^2+2 K[1]-y(x)}-\frac {1}{K[1]+1}\right )dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.681 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {\left (C_{1} \sqrt {x + 1} + 2 x + 2\right )^{2}}{4} - 1 \]