Internal problem ID [4183]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter 2, Equations of the first order and degree. page 20
Problem number: 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\frac {x}{y+1}-\frac {y y^{\prime }}{1+x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 720
dsolve(x/(1+y(x))=y(x)/(1+x)*diff(y(x),x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2} \\ y \relax (x ) = -\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{4}-\frac {1}{4 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}-\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{4}-\frac {1}{4 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}-\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1}+2 \sqrt {4 x^{6}+12 x^{5}+24 x^{3} c_{1}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.197 (sec). Leaf size: 346
DSolve[x/(1+y[x])==y[x]/(1+x)*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {1}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-1\right ) \\ y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\ y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right ) \\ \end{align*}