8.5.7 problem 7

Internal problem ID [735]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 7
Date solved : Wednesday, February 05, 2025 at 03:56:54 AM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 56

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

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

✓ Solution by Mathematica

Time used: 0.184 (sec). Leaf size: 63

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

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