Internal problem ID [15061]
Internal file name [OUTPUT/15062_Sunday_April_21_2024_01_22_00_PM_87799186/index.tex]
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.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\left (x -1\right ) {\mathrm e}^{-\frac {y^{2}}{2}}=-1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (x -1\right ) {\mathrm e}^{-\frac {y^{2}}{2}}=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-1+\left (x -1\right ) {\mathrm e}^{-\frac {y^{2}}{2}}}{y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5`[0, (exp(-1/2*y^2)*x-2*exp(-1/2*y^2)-1)/y]
✓ Solution by Maple
Time used: 0.0 (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 \left (x \right ) &= \sqrt {2}\, \sqrt {\ln \left (-c_{1} {\mathrm e}^{-x}+x -2\right )} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {\ln \left (-c_{1} {\mathrm e}^{-x}+x -2\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.375 (sec). Leaf size: 60
DSolve[y[x]*y'[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*}