20.4.9 problem Problem 17

Internal problem ID [3644]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 17
Date solved : Monday, January 27, 2025 at 07:49:37 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 3.534 (sec). Leaf size: 80

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

\[ y \left (x \right ) = \frac {x \left (-\operatorname {RootOf}\left (2 \textit {\_Z}^{6}+\left (9 c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-6 c_{1} \textit {\_Z}^{2} x^{2}+c_{1} x^{2}\right )^{2}+1\right )}{\operatorname {RootOf}\left (2 \textit {\_Z}^{6}+\left (9 c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-6 c_{1} \textit {\_Z}^{2} x^{2}+c_{1} x^{2}\right )^{2}} \]

✓ Solution by Mathematica

Time used: 60.196 (sec). Leaf size: 373

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

\begin{align*} y(x)\to \frac {\sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (-3 x^2+e^{2 c_1}\right )}{\sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-3 x^2+e^{2 c_1}\right )}{2^{2/3} \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-3 x^2+e^{2 c_1}\right )}{2^{2/3} \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ \end{align*}