Internal problem ID [2005]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 10, page 41
Problem number: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\[ \boxed {y^{2}+\left (2 y x -y^{2}\right ) y^{\prime }=-1} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 137
dsolve([(y(x)^2+1)+(2*x*y(x)-y(x)^2)*diff(y(x),x)=0,y(0) = -1],y(x), singsol=all)
\[ y = \frac {\left (-4+12 x +8 x^{3}+4 \sqrt {12 x^{4}-4 x^{3}+9 x^{2}-6 x +1}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4}-\frac {\left (i x \sqrt {3}+x -\left (-4+12 x +8 x^{3}+4 \sqrt {12 x^{4}-4 x^{3}+9 x^{2}-6 x +1}\right )^{\frac {1}{3}}\right ) x}{\left (-4+12 x +8 x^{3}+4 \sqrt {12 x^{4}-4 x^{3}+9 x^{2}-6 x +1}\right )^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 4.803 (sec). Leaf size: 100
DSolve[{(y[x]^2+1)+(2*x*y[x]-y[x]^2)*y'[x]==0,{y[0]==-1}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-2 x^3+\sqrt {12 x^4-4 x^3+9 x^2-6 x+1}-3 x+1}}-\frac {\sqrt [3]{-2 x^3+\sqrt {12 x^4-4 x^3+9 x^2-6 x+1}-3 x+1}}{\sqrt [3]{2}}+x \]