83.2.6 problem 6

Internal problem ID [19052]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (A) at page 8
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 12:48:16 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 119

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

\begin{align*} y \left (x \right ) &= \frac {\sqrt {-3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {\sqrt {-3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\sqrt {-3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\sqrt {-3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.244 (sec). Leaf size: 245

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

\begin{align*} y(x)\to -\sqrt {-3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-3 x^2-\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-3 x^2+\sqrt {8 x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-2 \sqrt {2} \sqrt {x^4}-3 x^2} \\ y(x)\to \sqrt {-2 \sqrt {2} \sqrt {x^4}-3 x^2} \\ y(x)\to -\sqrt {2 \sqrt {2} \sqrt {x^4}-3 x^2} \\ y(x)\to \sqrt {2 \sqrt {2} \sqrt {x^4}-3 x^2} \\ \end{align*}