Internal problem ID [1033]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number: 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
\[ \boxed {-2 y^{2}+\left (12 y^{2}-4 y x \right ) y^{\prime }=-2 x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 429
dsolve((2*x-2*y(x)^2)+(12*y(x)^2-4*x*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-6 x^{5}-6 c_{1} x^{3}+81 x^{4}+162 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{2}}{6 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-6 x^{5}-6 c_{1} x^{3}+81 x^{4}+162 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6} \\ y \left (x \right ) &= \frac {\left (-\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}+x \right ) \left (i \left (\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}+\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}-x \right )}{12 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{12}-\frac {x \left (i x \sqrt {3}+x -2 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}\right )}{12 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.8 (sec). Leaf size: 414
DSolve[(2*x-2*y[x]^2)+(12*y[x]^2-4*x*y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^2}{3\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}-\frac {\sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {x}{6} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) x^2}{6\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}+\frac {x}{6} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{6\ 2^{2/3} \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3+54 x^2+\sqrt {-4 x^6+4 \left (x^3-27 x^2-54 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}+\frac {x}{6} \\ \end{align*}