8.5.27 problem 27

Internal problem ID [755]
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
Date solved : Wednesday, February 05, 2025 at 03:58:15 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 55

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

\begin{align*} y &= {\left (\left (x^{3}+c_1 \right ) x \right )}^{{1}/{3}} \\ y &= -\frac {{\left (\left (x^{3}+c_1 \right ) x \right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {{\left (\left (x^{3}+c_1 \right ) x \right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.210 (sec). Leaf size: 72

DSolve[3*x*y[x]^2*D[y[x],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*}