Internal problem ID [12466]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 68.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Bernoulli]
\[ \boxed {3 y^{\prime } y^{2}-a y^{3}=x +1} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 106
dsolve(3*y(x)^2*diff(y(x),x)-a*y(x)^3-x-1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {{\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-1+\left (-x -1\right ) a \right ) a \right )}^{\frac {1}{3}}}{a} \\ y \left (x \right ) &= -\frac {{\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-1+\left (-x -1\right ) a \right ) a \right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 a} \\ y \left (x \right ) &= \frac {{\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-1+\left (-x -1\right ) a \right ) a \right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2 a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 17.994 (sec). Leaf size: 111
DSolve[3*y[x]^2*y'[x]-a*y[x]^3-x-1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{a^2 c_1 e^{a x}-a (x+1)-1}}{a^{2/3}} \\ \end{align*}