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]

\[ \boxed {3 y^{2} y^{\prime } x -y^{3}=3 x^{4}} \]

Solution by Maple

Time used: 0.0 (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 \left (x \right ) = \left (x^{4}+c_{1} x \right )^{\frac {1}{3}} y \left (x \right ) = -\frac {\left (x^{4}+c_{1} x \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (x^{4}+c_{1} x \right )^{\frac {1}{3}}}{2} y \left (x \right ) = -\frac {\left (x^{4}+c_{1} x \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (x^{4}+c_{1} x \right )^{\frac {1}{3}}}{2} \end{align*}

Solution by Mathematica

Time used: 0.217 (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*}