23.1.234 problem 230

Internal problem ID [4841]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 230
Date solved : Tuesday, September 30, 2025 at 08:43:58 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

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

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

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

✓ Mathematica. Time used: 0.299 (sec). Leaf size: 214

ode=(1+x)*D[y[x],x]==(1+y[x])+(1+x)*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: 0.689 (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)
 

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