Internal problem ID [6225]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.8. Integrating Factors. Page
32
Problem number: 1(a).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {\left (3 x^{2}-y^{2}\right ) y^{\prime }-2 x y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 317
dsolve((3*x^2-y(x)^2)*diff(y(x),x)-2*x*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}}{3 c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.184 (sec). Leaf size: 458
DSolve[(3*x^2-y[x]^2)*y'[x]-2*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right ) \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\ y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \\ \end{align*}