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 : Monday, January 27, 2025 at 11:08:14 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*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 6.190 (sec). Leaf size: 331

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

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