38.2.10 problem 10

Internal problem ID [6439]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 10
Date solved : Monday, January 27, 2025 at 02:03:26 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 23

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

\[ y = \operatorname {RootOf}\left (x c_1 \,\textit {\_Z}^{4}-c_1 x -\textit {\_Z} \right )^{2} x \]

✓ Solution by Mathematica

Time used: 60.158 (sec). Leaf size: 1659

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

\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}-3 \sqrt {\frac {8 x^2}{3}-\frac {16 \sqrt [3]{2} x^4}{3 \sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}-\frac {2 \sqrt {3} e^{c_1} x}{\sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}}-\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{3 \sqrt [3]{2}}}\right ) \\ y(x)\to \frac {1}{6} \left (3 \sqrt {\frac {8 x^2}{3}-\frac {16 \sqrt [3]{2} x^4}{3 \sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}-\frac {2 \sqrt {3} e^{c_1} x}{\sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}}-\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{3 \sqrt [3]{2}}}-\sqrt {3} \sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}-3 \sqrt {\frac {8 x^2}{3}-\frac {16 \sqrt [3]{2} x^4}{3 \sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {2 \sqrt {3} e^{c_1} x}{\sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}}-\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{3 \sqrt [3]{2}}}\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}+3 \sqrt {\frac {8 x^2}{3}-\frac {16 \sqrt [3]{2} x^4}{3 \sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {2 \sqrt {3} e^{c_1} x}{\sqrt {4 x^2+\frac {16 \sqrt [3]{2} x^4}{\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}+\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{\sqrt [3]{2}}}}-\frac {\sqrt [3]{128 x^6+27 e^{2 c_1} x^2+3 \sqrt {768 e^{2 c_1} x^8+81 e^{4 c_1} x^4}}}{3 \sqrt [3]{2}}}\right ) \\ \end{align*}