8.5.39 problem 39

Internal problem ID [767]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 39
Date solved : Monday, January 27, 2025 at 03:04:19 AM
CAS classification : [_exact, _rational]

\begin{align*} 3 x^{2} y^{3}+y^{4}+\left (3 x^{3} y^{2}+4 x y^{3}+y^{4}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 29

dsolve(3*x^2*y(x)^3+y(x)^4+(3*x^3*y(x)^2+4*x*y(x)^3+y(x)^4)*diff(y(x),x) = 0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ x y^{4}+x^{3} y^{3}+\frac {y^{5}}{5}+c_1 &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 30.987 (sec). Leaf size: 171

DSolve[3*x^2*y[x]^3+y[x]^4+(3*x^3*y[x]^2+4*x*y[x]^3+y[x]^4)*D[y[x],x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 0 \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}^4 x+5 \text {$\#$1}^3 x^3-5 c_1\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}^4 x+5 \text {$\#$1}^3 x^3-5 c_1\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}^4 x+5 \text {$\#$1}^3 x^3-5 c_1\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}^4 x+5 \text {$\#$1}^3 x^3-5 c_1\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^5+5 \text {$\#$1}^4 x+5 \text {$\#$1}^3 x^3-5 c_1\&,5\right ] \\ y(x)\to 0 \\ \end{align*}