20.5.7 problem Problem 7

Internal problem ID [3690]
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
Date solved : Monday, January 27, 2025 at 07:55:25 AM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} 4 \,{\mathrm e}^{2 x}+2 y x -y^{2}+\left (x -y\right )^{2} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 115

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 \left (x \right ) &= \left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1} \right )^{{1}/{3}}+x \\ y \left (x \right ) &= -\frac {\left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1} \right )^{{1}/{3}}}{2}-\frac {i \sqrt {3}\, \left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1} \right )^{{1}/{3}}}{2}+x \\ y \left (x \right ) &= -\frac {\left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1} \right )^{{1}/{3}}}{2}+\frac {i \sqrt {3}\, \left (-x^{3}-6 \,{\mathrm e}^{2 x}-3 c_{1} \right )^{{1}/{3}}}{2}+x \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.433 (sec). Leaf size: 112

DSolve[(4*Exp[2*x]+2*x*y[x]-y[x]^2)+(x-y[x])^2*D[y[x],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*}