Internal problem ID [3456]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 718.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {\left (1+a \left (x +y\right )\right )^{n} y^{\prime }+a \left (x +y\right )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 45
dsolve((1+a*(x+y(x)))^n*diff(y(x),x)+a*(x+y(x))^n = 0,y(x), singsol=all)
\[ y \relax (x ) = -x +\RootOf \left (-x -\left (\int _{}^{\textit {\_Z}}-\frac {\left (a \textit {\_a} +1\right )^{n}}{-a \,\textit {\_a}^{n}+\left (a \textit {\_a} +1\right )^{n}}d \textit {\_a} \right )+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 5.565 (sec). Leaf size: 331
DSolve[(1+a (x+y[x]))^n y'[x]+a(x+y[x])^n==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {a (K[1]+y(x))^n}{a (K[1]+y(x))^n-(a (K[1]+y(x))+1)^n}dK[1]+\int _1^{y(x)}-\frac {-a \int _1^x\left (\frac {a n (K[1]+K[2])^{n-1}}{a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n}-\frac {a (K[1]+K[2])^n \left (a n (K[1]+K[2])^{n-1}-a n (a (K[1]+K[2])+1)^{n-1}\right )}{\left (a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n\right )^2}\right )dK[1] (x+K[2])^n+(a (x+K[2])+1)^n+(a (x+K[2])+1)^n \int _1^x\left (\frac {a n (K[1]+K[2])^{n-1}}{a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n}-\frac {a (K[1]+K[2])^n \left (a n (K[1]+K[2])^{n-1}-a n (a (K[1]+K[2])+1)^{n-1}\right )}{\left (a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n\right )^2}\right )dK[1]}{(a (x+K[2])+1)^n-a (x+K[2])^n}dK[2]=c_1,y(x)\right ] \]