29.25.21 problem 718
Internal
problem
ID
[5308]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
718
Date
solved
:
Sunday, March 30, 2025 at 07:53:10 AM
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: 6.19 (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: 29.508 (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)
\[
y{\left (x \right )} = - 2^{n} a \int \limits ^{- C_{2} - x} \frac {\left (- r\right )^{n}}{2^{n} a \left (- r\right )^{n} - \left (- 2 r a + 2\right )^{n}}\, dr + C_{1} + x - \int \limits ^{- C_{2} - x} \frac {\left (- 2 r a + 2\right )^{n}}{2^{n} a \left (- r\right )^{n} - \left (- 2 r a + 2\right )^{n}}\, dr
\]