75.12.17 problem 291

Internal problem ID [16883]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 291
Date solved : Tuesday, January 28, 2025 at 09:39:28 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Bernoulli]

\begin{align*} x y^{2} y^{\prime }-y^{3}&=\frac {x^{4}}{3} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 47

dsolve(x*y(x)^2*diff(y(x),x)-y(x)^3=1/3*x^4,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.187 (sec). Leaf size: 54

DSolve[x*y[x]^2*D[y[x],x]-y[x]^3==1/3*x^4,y[x],x,IncludeSingularSolutions -> True]
 

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