Internal problem ID [4472]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson
10
Problem number: Recognizable Exact Differential equations. Integrating factors. Example
10.741, page 90.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class C`]]
\[ \boxed {y^{3}+y^{2} x +y+\left (x^{3}+x^{2} y+x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve((y(x)^3+x*y(x)^2+y(x))+(x^3+x^2*y(x)+x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {x^{4}+2 x^{2}+1}{x \left (\sqrt {\frac {c_{1} x^{4}+c_{1} x^{2}-1}{x^{2} \left (x^{2}+1\right )}}\, \left (x^{2}+1\right )^{\frac {3}{2}}-x^{2}-1\right )} y \left (x \right ) = -\frac {x^{4}+2 x^{2}+1}{x \left (x^{2}+\sqrt {\frac {c_{1} x^{4}+c_{1} x^{2}-1}{x^{2} \left (x^{2}+1\right )}}\, \left (x^{2}+1\right )^{\frac {3}{2}}+1\right )} \end{align*}
✓ Solution by Mathematica
Time used: 3.726 (sec). Leaf size: 114
DSolve[(y[x]^3+x*y[x]^2+y[x])+(x^3+x^2*y[x]+x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {1}{x^3}} x \left (x^2+1\right )}{\sqrt {\frac {1}{x^3}} x^2-\sqrt {c_1 x^3-\frac {1}{x}+c_1 x}} y(x)\to -\frac {\sqrt {\frac {1}{x^3}} x \left (x^2+1\right )}{\sqrt {\frac {1}{x^3}} x^2+\sqrt {c_1 x^3-\frac {1}{x}+c_1 x}} y(x)\to 0 \end{align*}