Internal problem ID [15194]
Internal file name [OUTPUT/15195_Tuesday_April_23_2024_04_53_53_PM_67792014/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 14. Differential equations admitting of
depression of their order. Exercises page 107
Problem number: 337.
ODE order: 1.
ODE degree: 0.
The type(s) of ODE detected by this program : "separable"
Maple gives the following as the ode type
[_separable]
\[ \boxed {y x -y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right )=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y x}{\operatorname {LambertW}\left (y \right )} \end {align*}
Where \(f(x)=x\) and \(g(y)=\frac {y}{\operatorname {LambertW}\left (y \right )}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y}{\operatorname {LambertW}\left (y \right )}} \,dy &= x \,d x \\ \int { \frac {1}{\frac {y}{\operatorname {LambertW}\left (y \right )}} \,dy} &= \int {x \,d x} \\ \frac {\operatorname {LambertW}\left (y \right )^{2}}{2}+\operatorname {LambertW}\left (y \right )&=\frac {x^{2}}{2}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \left (-1+\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1+\sqrt {x^{2}+2 c_{1} +1}} \\ y &= \left (-1-\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1-\sqrt {x^{2}+2 c_{1} +1}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \left (-1+\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1+\sqrt {x^{2}+2 c_{1} +1}} \\ \tag{2} y &= \left (-1-\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1-\sqrt {x^{2}+2 c_{1} +1}} \\ \end{align*}
Verification of solutions
\[ y = \left (-1+\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1+\sqrt {x^{2}+2 c_{1} +1}} \] Verified OK.
\[ y = \left (-1-\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1-\sqrt {x^{2}+2 c_{1} +1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y x -y^{\prime } \ln \left (\frac {y^{\prime }}{x}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y x}{\mathit {LambertW}\left (y\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \mathit {LambertW}\left (y\right )}{y}=x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \mathit {LambertW}\left (y\right )}{y}d x =\int x d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\mathit {LambertW}\left (y\right )^{2}}{2}+\mathit {LambertW}\left (y\right )=\frac {x^{2}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\left (-1-\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1-\sqrt {x^{2}+2 c_{1} +1}}, y=\left (-1+\sqrt {x^{2}+2 c_{1} +1}\right ) {\mathrm e}^{-1+\sqrt {x^{2}+2 c_{1} +1}}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying dAlembert trying simple symmetries for implicit equations <- symmetries for implicit equations successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 63
dsolve(x*y(x)=diff(y(x),x)*ln(diff(y(x),x)/x),y(x), singsol=all)
\begin{align*} y &= \left (-1-\sqrt {x^{2}-2 c_{1} +1}\right ) {\mathrm e}^{-1-\sqrt {x^{2}-2 c_{1} +1}} \\ y &= \left (-1+\sqrt {x^{2}-2 c_{1} +1}\right ) {\mathrm e}^{-1+\sqrt {x^{2}-2 c_{1} +1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.223 (sec). Leaf size: 83
DSolve[x*y[x]==y'[x]*Log[y'[x]/x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{-1-\sqrt {x^2+1+2 c_1}} \left (1+\sqrt {x^2+1+2 c_1}\right ) \\ y(x)\to e^{-1+\sqrt {x^2+1+2 c_1}} \left (-1+\sqrt {x^2+1+2 c_1}\right ) \\ y(x)\to 0 \\ \end{align*}