60.1.288 problem 289

Internal problem ID [10302]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 289
Date solved : Monday, January 27, 2025 at 06:51:59 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} \left (6 y-x \right )^{2} y^{\prime }-6 y^{2}+2 y x +a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 113

dsolve((6*y(x)-x)^2*diff(y(x),x)-6*y(x)^2+2*x*y(x)+a=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{6}+\frac {x}{6} \\ y &= -\frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}-\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}+\frac {x}{6} \\ y &= -\frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}+\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}+\frac {x}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.689 (sec). Leaf size: 115

DSolve[(6*y[x]-x)^2*D[y[x],x]-6*y[x]^2+2*x*y[x]+a==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{6} \left (x+\sqrt [3]{-18 a x-x^3+18 c_1}\right ) \\ y(x)\to \frac {x}{6}+\frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-18 a x-x^3+18 c_1} \\ y(x)\to \frac {x}{6}-\frac {1}{12} \left (1+i \sqrt {3}\right ) \sqrt [3]{-18 a x-x^3+18 c_1} \\ \end{align*}