Internal problem ID [3964]
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]
\[ \boxed {\left (1+a \left (y+x \right )\right )^{n} y^{\prime }+a \left (y+x \right )^{n}=0} \]
✓ Solution by Maple
Time used: 0.109 (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: 7.033 (sec). Leaf size: 331
DSolve[(1+a*(x+y[x]))^n*y'[x]+a*(x+y[x])^n==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {a (K[1]+y(x))^n}{a (K[1]+y(x))^n-(a (K[1]+y(x))+1)^n}dK[1]+\int _1^{y(x)}-\frac {-a \int _1^x\left (\frac {a n (K[1]+K[2])^{n-1}}{a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n}-\frac {a (K[1]+K[2])^n \left (a n (K[1]+K[2])^{n-1}-a n (a (K[1]+K[2])+1)^{n-1}\right )}{\left (a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n\right )^2}\right )dK[1] (x+K[2])^n+(a (x+K[2])+1)^n+(a (x+K[2])+1)^n \int _1^x\left (\frac {a n (K[1]+K[2])^{n-1}}{a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n}-\frac {a (K[1]+K[2])^n \left (a n (K[1]+K[2])^{n-1}-a n (a (K[1]+K[2])+1)^{n-1}\right )}{\left (a (K[1]+K[2])^n-(a (K[1]+K[2])+1)^n\right )^2}\right )dK[1]}{(a (x+K[2])+1)^n-a (x+K[2])^n}dK[2]=c_1,y(x)\right ] \]