12.5.45 problem 44

Internal problem ID [1669]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 44
Date solved : Monday, January 27, 2025 at 05:23:59 AM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.184 (sec). Leaf size: 69

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

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