29.22.17 problem 625

Internal problem ID [5217]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 22
Problem number : 625
Date solved : Monday, January 27, 2025 at 10:22:48 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 104

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

\begin{align*} y \left (x \right ) &= \frac {{\left (a \left (c_{1} {\mathrm e}^{a x} a^{2}-1+\left (-x -1\right ) a \right )\right )}^{{1}/{3}}}{a} \\ y \left (x \right ) &= -\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 \left (x \right ) &= \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: 24.430 (sec). Leaf size: 111

DSolve[3 y[x]^2 D[y[x],x]==1+x+a y[x]^3,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*}