75.11.13 problem 272

Internal problem ID [16864]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 11. Singular solutions of differential equations. Exercises page 92
Problem number : 272
Date solved : Tuesday, January 28, 2025 at 09:35:44 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}-y^{\prime } y+{\mathrm e}^{x}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.783 (sec). Leaf size: 34

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

\begin{align*} y &= -2 \,{\mathrm e}^{\frac {x}{2}} \\ y &= 2 \,{\mathrm e}^{\frac {x}{2}} \\ y &= \frac {c_{1}^{2} {\mathrm e}^{x}+1}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.885 (sec). Leaf size: 163

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

\begin{align*} y(x)\to -\frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {x-c_1}{2}\right )}} \\ y(x)\to \frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {x-c_1}{2}\right )}} \\ y(x)\to -\frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {-x+c_1}{2}\right )}} \\ y(x)\to \frac {2 i e^{x/2}}{\sqrt {-1+\tanh ^2\left (\frac {-x+c_1}{2}\right )}} \\ y(x)\to -2 e^{x/2} \\ y(x)\to 2 e^{x/2} \\ \end{align*}