Internal problem ID [2210]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 1, First-Order Differential Equations. Section 1.9, Exact Differential Equations. page
91
Problem number: Problem 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, [_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {4 \,{\mathrm e}^{2 x}+2 y x -y^{2}+\left (x -y\right )^{2} y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 117
dsolve((4*exp(2*x)+2*x*y(x)-y(x)^2)+(x-y(x))^2*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1}\right )^{\frac {1}{3}}+x \\ y \relax (x ) = -\frac {\left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1}\right )^{\frac {1}{3}}}{2}+x \\ y \relax (x ) = -\frac {\left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1}\right )^{\frac {1}{3}}}{2}+x \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.511 (sec). Leaf size: 112
DSolve[(4*Exp[2*x]+2*x*y[x]-y[x]^2)+(x-y[x])^2*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x+\sqrt [3]{-x^3-6 e^{2 x}+3 c_1} \\ y(x)\to x+\frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^3-6 e^{2 x}+3 c_1} \\ y(x)\to x-\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3-6 e^{2 x}+3 c_1} \\ \end{align*}