62.7.5 problem Ex 5

Internal problem ID [12749]
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 : Wednesday, March 05, 2025 at 08:24:22 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.010 (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);
 

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

✓ Mathematica. Time used: 0.514 (sec). Leaf size: 214

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

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

✓ Sympy. Time used: 1.016 (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)
 

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