Internal problem ID [15059]
Internal file name [OUTPUT/15060_Sunday_April_21_2024_01_21_56_PM_98789621/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: 167.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-y \left ({\mathrm e}^{x}+\ln \left (y\right )\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \left ({\mathrm e}^{x}+\ln \left (y\right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \left ({\mathrm e}^{x}+\ln \left (y\right )\right ) \end {array} \]
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, y*exp(x)], [0, -exp(x)*x*y+y*ln(y)]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 11
dsolve(diff(y(x),x)=y(x)*(exp(x)+ln(y(x))),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{{\mathrm e}^{x} \left (x +c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 0.372 (sec). Leaf size: 15
DSolve[y'[x]==y[x]*(Exp[x]+Log[y[x]]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{e^x (x+c_1)} \]