Internal problem ID [10150]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 17. Other forms
which Integrating factors can be found. Page 25
Problem number: Ex 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {y^{4}+2 y+\left (x y^{3}+2 y^{4}-4 x \right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve((y(x)^4+2*y(x))+(x*y(x)^3+2*y(x)^4-4*x)*diff(y(x),x)=0,y(x), singsol=all)
\[ x -\frac {\left (-y \relax (x )^{2}+c_{1}\right ) y \relax (x )^{2}}{y \relax (x )^{3}+2} = 0 \]
✓ Solution by Mathematica
Time used: 60.196 (sec). Leaf size: 2021
DSolve[(y[x]^4+2*y[x])+(x*y[x]^3+2*y[x]^4-4*x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}+\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}+\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}-\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ y(x)\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{2}-\frac {x \left (x^2+4 c_1\right )}{4 \sqrt {\frac {\sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (24 x+c_1{}^2\right )}{3 \sqrt [3]{54 x^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (24 x+c_1{}^2\right ){}^3}+144 c_1 x-2 c_1{}^3}}+\frac {x^2}{4}+\frac {2 c_1}{3}}}+\frac {4 c_1}{3}}-\frac {x}{4} \\ \end{align*}