Internal problem ID [2334]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 37, page 171
Problem number: 21.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
\[ \boxed {y {y^{\prime }}^{2}+2 y^{\prime }=-1} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 53
dsolve(diff(y(x),x)^2*y(x)+2*diff(y(x),x)+1=0,y(x), singsol=all)
\begin{align*} \frac {\left (2 y \left (x \right )-2\right ) \sqrt {1-y \left (x \right )}}{3}+x -c_{1} +y \left (x \right )-1 &= 0 \\ \frac {\left (-2 y \left (x \right )+2\right ) \sqrt {1-y \left (x \right )}}{3}+x -c_{1} +y \left (x \right )-1 &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.957 (sec). Leaf size: 1098
DSolve[y'[x]^2*y[x]+2*y'[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{3} \left (-24 x^2+36 x-48 c_1 x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1\right ){}^{2/3}+\sqrt [3]{-24 x^2+36 x-48 c_1 x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1}-8\ 3^{2/3} x+3\ 3^{2/3}-8\ 3^{2/3} c_1}{4 \sqrt [3]{-24 x^2+(36-48 c_1) x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1}} \\ y(x)\to \frac {1}{8} \left (\frac {3^{2/3} \left (1+i \sqrt {3}\right ) (8 x-3+8 c_1)}{\sqrt [3]{-24 x^2+(36-48 c_1) x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1}}+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \sqrt [3]{-24 x^2+36 x-48 c_1 x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1}+2\right ) \\ y(x)\to \frac {1}{8} \left (\frac {3^{2/3} \left (1-i \sqrt {3}\right ) (8 x-3+8 c_1)}{\sqrt [3]{-24 x^2+(36-48 c_1) x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1}}-\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-24 x^2+36 x-48 c_1 x+8 \sqrt {3} \sqrt {(x+c_1){}^3 (3 x-1+3 c_1)}-9-24 c_1{}^2+36 c_1}+2\right ) \\ y(x)\to \frac {\sqrt [3]{3} \left (-24 x^2+36 x+48 c_1 x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1\right ){}^{2/3}+\sqrt [3]{-24 x^2+36 x+48 c_1 x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1}-8\ 3^{2/3} x+3\ 3^{2/3}+8\ 3^{2/3} c_1}{4 \sqrt [3]{-24 x^2+12 (3+4 c_1) x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1}} \\ y(x)\to \frac {1}{8} \left (\frac {3^{2/3} \left (1+i \sqrt {3}\right ) (8 x-3-8 c_1)}{\sqrt [3]{-24 x^2+12 (3+4 c_1) x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1}}+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \sqrt [3]{-24 x^2+12 (3+4 c_1) x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1}+2\right ) \\ y(x)\to \frac {1}{8} \left (\frac {3^{2/3} \left (1-i \sqrt {3}\right ) (8 x-3-8 c_1)}{\sqrt [3]{-24 x^2+12 (3+4 c_1) x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1}}-\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-24 x^2+12 (3+4 c_1) x+8 \sqrt {3} \sqrt {(-x+c_1){}^3 (-3 x+1+3 c_1)}-9-24 c_1{}^2-36 c_1}+2\right ) \\ \end{align*}