23.1.725 problem 722

Internal problem ID [5332]
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 : 722
Date solved : Tuesday, September 30, 2025 at 12:30:33 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} \left (1+a \left (x +y\right )\right )^{n} y^{\prime }+a \left (x +y\right )^{n}&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 42

ode:=(1+a*(x+y(x)))^n*diff(y(x),x)+a*(x+y(x))^n = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} a +1\right )^{n}}{-a \,\textit {\_a}^{n}+\left (\textit {\_a} a +1\right )^{n}}d \textit {\_a} +c_1 \right ) \]

✓ Mathematica. Time used: 0.736 (sec). Leaf size: 331

ode=(1+a*(x+y[x]))^n*D[y[x],x]+a*(x+y[x])^n==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 41.640 (sec). Leaf size: 71

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

\[ y{\left (x \right )} = - 2^{n} a \int \limits ^{- C_{2} - x} \frac {\left (- r\right )^{n}}{2^{n} a \left (- r\right )^{n} - \left (- 2 r a + 2\right )^{n}}\, dr + C_{1} + x - \int \limits ^{- C_{2} - x} \frac {\left (- 2 r a + 2\right )^{n}}{2^{n} a \left (- r\right )^{n} - \left (- 2 r a + 2\right )^{n}}\, dr \]