2.5.3 Problem 4

Internal problem ID [18552]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 4
Date solved : Saturday, February 22, 2025 at 09:27:10 PM
CAS classification : [_quadrature]

Solve

\begin{align*} y^{\prime }&={\mathrm e}^{z -y^{\prime }} \end{align*}

Solved as first order quadrature ode

Time used: 0.084 (sec)

Since the ode has the form \(y^{\prime }=f(z)\), then we only need to integrate \(f(z)\).

\begin{align*} \int {dy} &= \int {\operatorname {LambertW}\left ({\mathrm e}^{z}\right )\, dz}\\ y &= \operatorname {LambertW}\left ({\mathrm e}^{z}\right )+\frac {\operatorname {LambertW}\left ({\mathrm e}^{z}\right )^{2}}{2} + c_1 \end{align*}

Figure 2.51: Slope field \(y^{\prime } = {\mathrm e}^{z -y^{\prime }}\)

Summary of solutions found

\begin{align*} y &= \operatorname {LambertW}\left ({\mathrm e}^{z}\right )+\frac {\operatorname {LambertW}\left ({\mathrm e}^{z}\right )^{2}}{2}+c_1 \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }={\mathrm e}^{z -y^{\prime }} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\mathit {LambertW}\left ({\mathrm e}^{z}\right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} z \\ {} & {} & \int y^{\prime }d z =\int \mathit {LambertW}\left ({\mathrm e}^{z}\right )d z +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\mathit {LambertW}\left ({\mathrm e}^{z}\right )^{2}}{2}+\mathit {LambertW}\left ({\mathrm e}^{z}\right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\mathit {LambertW}\left ({\mathrm e}^{z}\right )^{2}}{2}+\mathit {LambertW}\left ({\mathrm e}^{z}\right )+\mathit {C1} \end {array} \]

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 
<- differential order: 1; missing  y(x)  successful`
 

Maple dsolve solution

Solving time : 0.004 (sec)
Leaf size : 16

dsolve(diff(y(z),z) = exp(z-diff(y(z),z)),y(z),singsol=all)
 

\[ y = \frac {\operatorname {LambertW}\left ({\mathrm e}^{z}\right )^{2}}{2}+\operatorname {LambertW}\left ({\mathrm e}^{z}\right )+c_{1} \]

✓Mathematica DSolve solution

Solving time : 0.023 (sec)
Leaf size : 22

DSolve[{D[y[z],z]==Exp[z-D[y[z],z]],{}},y[z],z,IncludeSingularSolutions->True]
 

\[ y(z)\to \frac {1}{2} W\left (e^z\right )^2+W\left (e^z\right )+c_1 \]

✓Sympy solution

Solving time : 0.253 (sec)
Leaf size : 17

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq(-exp(z - Derivative(y(z), z)) + Derivative(y(z), z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics)
 

Eq(y(z), C1 + LambertW(exp(z))**2/2 + LambertW(exp(z)))
 

\[ y{\left (z \right )} = C_{1} + \frac {W^{2}\left (e^{z}\right )}{2} + W\left (e^{z}\right ) \]