Internal problem ID [6065]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 5. Existence and uniqueness of solutions to first order equations. Page
190
Problem number: 1(c).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime }-\frac {x^{2}+x}{y-y^{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 498
dsolve(diff(y(x),x)=(x+x^2)/(y(x)-y(x)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (1-4 x^{3}-6 x^{2}-12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+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 c_{1} x^{3}+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 \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {2}{3}}-i \sqrt {3}-2 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}+1}{4 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {2}{3}}-i \sqrt {3}+2 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}-1}{4 \left (-4 x^{3}-6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}-12 c_{1} +1\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.147 (sec). Leaf size: 346
DSolve[y'[x]==(x+x^2)/(y[x]-y[x]^2),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*}