5.27 problem 27

Internal problem ID [105]

Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.6, Substitution methods and exact equations. Page 74
Problem number: 27.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class G], _rational, _Bernoulli]

Solve \begin {gather*} \boxed {3 x y^{2} y^{\prime }-3 x^{4}-y^{3}=0} \end {gather*}

Solution by Maple

Time used: 0.004 (sec). Leaf size: 73

dsolve(3*x*y(x)^2*diff(y(x),x) = 3*x^4+y(x)^3,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \left (x^{4}+x c_{1}\right )^{\frac {1}{3}} \\ y \relax (x ) = -\frac {\left (x^{4}+x c_{1}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (x^{4}+x c_{1}\right )^{\frac {1}{3}}}{2} \\ y \relax (x ) = -\frac {\left (x^{4}+x c_{1}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (x^{4}+x c_{1}\right )^{\frac {1}{3}}}{2} \\ \end{align*}

Solution by Mathematica

Time used: 0.175 (sec). Leaf size: 72

DSolve[3*x*y[x]^2*y'[x] == 3*x^4+y[x]^3,y[x],x,IncludeSingularSolutions -> True]
 

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