56.7.5 problem Ex 5

Internal problem ID [14109]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 14. Equations reducible to linear equations (Bernoulli). Page 21
Problem number : Ex 5
Date solved : Thursday, October 02, 2025 at 09:14:25 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{\prime }-\frac {1+y}{1+x}&=\sqrt {1+y} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 81

ode:=diff(y(x),x)-(1+y(x))/(1+x) = (1+y(x))^(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: 0.224 (sec). Leaf size: 60

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

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

✓ Sympy. Time used: 0.676 (sec). Leaf size: 20

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

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