83.22.22 problem 22

Internal problem ID [19276]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 22
Date solved : Tuesday, January 28, 2025 at 01:22:06 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}-\left (y+2 x -{\mathrm e}^{x -y}\right ) {y^{\prime }}^{2}+\left (2 y x -2 x \,{\mathrm e}^{x -y}-y \,{\mathrm e}^{x -y}\right ) y^{\prime }+2 x y \,{\mathrm e}^{x -y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 28

dsolve(diff(y(x),x)^3-(y(x)+2*x-exp(x-y(x)))*diff(y(x),x)^2+(2*x*y(x)-2*x*exp(x-y(x))-y(x)*exp(x-y(x)))*diff(y(x),x)+2*x*y(x)*exp(x-y(x))=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{x} c_{1} \\ y \left (x \right ) &= x^{2}+c_{1} \\ y \left (x \right ) &= \ln \left (-{\mathrm e}^{x}-c_{1} \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.023 (sec). Leaf size: 39

DSolve[D[y[x],x]^3-(y[x]+2*x-Exp[x-y[x]])*D[y[x],x]^2+(2*x*y[x]-2*x*Exp[x-y[x]]-y[x]*Exp[x-y[x]])*D[y[x],x]+2*x*y[x]*Exp[x-y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 e^x \\ y(x)\to x^2+c_1 \\ y(x)\to \log \left (-e^x+c_1\right ) \\ y(x)\to 0 \\ \end{align*}