Internal problem ID [103]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.6, Substitution methods and exact equations. Page 74
Problem number: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
Solve \begin {gather*} \boxed {\sqrt {x^{4}+1}\, y^{2} \left (y+y^{\prime } x \right )-x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 135
dsolve((x^4+1)^(1/2)*y(x)^2*(y(x)+x*diff(y(x),x)) = x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (3 \left (\int \frac {x^{3}}{\sqrt {x^{4}+1}}d x \right )+c_{1}\right )^{\frac {1}{3}}}{x} \\ y \relax (x ) = \frac {-\frac {\left (3 \left (\int \frac {x^{3}}{\sqrt {x^{4}+1}}d x \right )+c_{1}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (3 \left (\int \frac {x^{3}}{\sqrt {x^{4}+1}}d x \right )+c_{1}\right )^{\frac {1}{3}}}{2}}{x} \\ y \relax (x ) = \frac {-\frac {\left (3 \left (\int \frac {x^{3}}{\sqrt {x^{4}+1}}d x \right )+c_{1}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (3 \left (\int \frac {x^{3}}{\sqrt {x^{4}+1}}d x \right )+c_{1}\right )^{\frac {1}{3}}}{2}}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.843 (sec). Leaf size: 106
DSolve[(x^4+1)^(1/2)*y[x]^2*(y[x]+x*y'[x]) ==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [3]{\frac {3 \sqrt {x^4+1}}{2 x^3}+\frac {c_1}{x^3}} \\ y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{\frac {3 \sqrt {x^4+1}+2 c_1}{x^3}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{\frac {3 \sqrt {x^4+1}}{2 x^3}+\frac {c_1}{x^3}} \\ \end{align*}