69.1.49 problem 68

Internal problem ID [14202]
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
Date solved : Tuesday, January 28, 2025 at 06:21:42 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 3 y^{2} y^{\prime }-a y^{3}-x -1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 104

dsolve(3*y(x)^2*diff(y(x),x)-a*y(x)^3-x-1=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}}}{a} \\ y &= -\frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 a} \\ y &= \frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.361 (sec). Leaf size: 144

DSolve[3*y[x]^2*D[y[x],x]-a*y[x]^3-x-1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{\frac {a x}{3}} \sqrt [3]{3 \int _1^x\frac {1}{3} e^{-a K[1]} (K[1]+1)dK[1]+c_1} \\ y(x)\to -\sqrt [3]{-1} e^{\frac {a x}{3}} \sqrt [3]{3 \int _1^x\frac {1}{3} e^{-a K[1]} (K[1]+1)dK[1]+c_1} \\ y(x)\to (-1)^{2/3} e^{\frac {a x}{3}} \sqrt [3]{3 \int _1^x\frac {1}{3} e^{-a K[1]} (K[1]+1)dK[1]+c_1} \\ \end{align*}