Internal problem ID [2703]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 69.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {1+y+\left (x -y \left (1+y\right )^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 33
dsolve((1+y(x))+(x-y(x)*(1+y(x))^2)* diff(y(x),x)=0,y(x), singsol=all)
\[ x -\frac {\frac {y \relax (x )^{4}}{4}+\frac {2 y \relax (x )^{3}}{3}+\frac {y \relax (x )^{2}}{2}+c_{1}}{y \relax (x )+1} = 0 \]
✓ Solution by Mathematica
Time used: 38.302 (sec). Leaf size: 1586
DSolve[(1+y[x])+(x-y[x]*(1+y[x])^2)* y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}-3 \sqrt {-\frac {8 (27 x+2)}{9 \sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}}-\frac {2}{3} \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+\frac {2 (4 x-1-12 c_1)}{3 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+\frac {8}{9}}-4\right ) \\ y(x)\to \frac {1}{6} \left (-\sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}+3 \sqrt {-\frac {8 (27 x+2)}{9 \sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}}-\frac {2}{3} \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+\frac {2 (4 x-1-12 c_1)}{3 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+\frac {8}{9}}-4\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}-3 \sqrt {\frac {8 (27 x+2)}{9 \sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}}-\frac {2}{3} \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+\frac {2 (4 x-1-12 c_1)}{3 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+\frac {8}{9}}-4\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}+3 \sqrt {\frac {8 (27 x+2)}{9 \sqrt {\frac {-24 x+6+72 c_1}{\sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+6 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+4}}-\frac {2}{3} \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}+\frac {2 (4 x-1-12 c_1)}{3 \sqrt [3]{27 x^2-\frac {1}{432} \sqrt {186624 \left (27 x^2+1+12 c_1\right ){}^2-4 (-144 x+36+432 c_1){}^3}+1+12 c_1}}+\frac {8}{9}}-4\right ) \\ y(x)\to -1 \\ \end{align*}