75.5.18 problem 117

Internal problem ID [16754]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 5. Homogeneous equations. Exercises page 44
Problem number : 117
Date solved : Tuesday, January 28, 2025 at 09:21:15 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} 4 y^{6}+x^{3}&=6 x y^{5} y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 123

dsolve(4*y(x)^6+x^3=6*x*y(x)^5*diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y &= \left (x^{3} \left (c_{1} x -1\right )\right )^{{1}/{6}} \\ y &= -\left (x^{3} \left (c_{1} x -1\right )\right )^{{1}/{6}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (x^{3} \left (c_{1} x -1\right )\right )^{{1}/{6}}}{2} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (x^{3} \left (c_{1} x -1\right )\right )^{{1}/{6}}}{2} \\ y &= -\frac {\left (i \sqrt {3}-1\right ) \left (x^{3} \left (c_{1} x -1\right )\right )^{{1}/{6}}}{2} \\ y &= \frac {\left (1+i \sqrt {3}\right ) \left (x^{3} \left (c_{1} x -1\right )\right )^{{1}/{6}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.432 (sec). Leaf size: 144

DSolve[4*y[x]^6+x^3==6*x*y[x]^5*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {x} \sqrt [6]{-1+c_1 x} \\ y(x)\to \sqrt {x} \sqrt [6]{-1+c_1 x} \\ y(x)\to -\sqrt [3]{-1} \sqrt {x} \sqrt [6]{-1+c_1 x} \\ y(x)\to \sqrt [3]{-1} \sqrt {x} \sqrt [6]{-1+c_1 x} \\ y(x)\to -(-1)^{2/3} \sqrt {x} \sqrt [6]{-1+c_1 x} \\ y(x)\to (-1)^{2/3} \sqrt {x} \sqrt [6]{-1+c_1 x} \\ \end{align*}