Internal problem ID [3212]
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]`]]
\[ \boxed {y+\left (x -y \left (y+1\right )^{2}\right ) y^{\prime }=-1} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve((1+y(x))+(x-y(x)*(1+y(x))^2)* diff(y(x),x)=0,y(x), singsol=all)
\[ x +\frac {-3 y \left (x \right )^{4}-8 y \left (x \right )^{3}-6 y \left (x \right )^{2}-12 c_{1}}{12 y \left (x \right )+12} = 0 \]
✓ Solution by Mathematica
Time used: 33.714 (sec). Leaf size: 1594
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 {3 \left (32 x+\frac {64}{27}\right )}{4 \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 {3 \left (32 x+\frac {64}{27}\right )}{4 \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 {3 \left (32 x+\frac {64}{27}\right )}{4 \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 {3 \left (32 x+\frac {64}{27}\right )}{4 \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*}