75.6.36 problem 169

Internal problem ID [16792]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 169
Date solved : Tuesday, January 28, 2025 at 08:26:18 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 42

dsolve(y(x)*diff(y(x),x)+1=(x-1)*exp(-y(x)^2/2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 7.419 (sec). Leaf size: 60

DSolve[y[x]*D[y[x],x]+1==(x-1)*Exp[-y[x]^2/2],y[x],x,IncludeSingularSolutions -> True]
 

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