77.1.79 problem 98 (page 135)

Internal problem ID [17969]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 98 (page 135)
Date solved : Tuesday, January 28, 2025 at 11:19:00 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: 2.173 (sec). Leaf size: 34

dsolve(diff(y(x),x)^2-diff(y(x),x)*y(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: 7.257 (sec). Leaf size: 163

DSolve[D[y[x],x]^2-D[y[x],x]*y[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*}