69.4.9 problem 54
Internal
problem
ID
[17998]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
4.
Equations
with
variables
separable
and
equations
reducible
to
them.
Exercises
page
38
Problem
number
:
54
Date
solved
:
Thursday, October 02, 2025 at 02:32:46 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=a^{x +y} \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 22
ode:=diff(y(x),x) = a^(x+y(x));
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\ln \left (-\frac {1}{c_1 \ln \left (a \right )+a^{x}}\right )}{\ln \left (a \right )}
\]
✓ Mathematica. Time used: 3.577 (sec). Leaf size: 24
ode=D[y[x],x]==a^(x+y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\log \left (-a^x-c_1 \log (a)\right )}{\log (a)} \end{align*}
✓ Sympy. Time used: 3.361 (sec). Leaf size: 144
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**(x + y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \begin {cases} \frac {\log {\left (- C_{1} x - \frac {C_{1}}{\log {\left (a \right )}} \right )}}{\log {\left (a \right )}} & \text {for}\: C_{1} x \log {\left (a \right )} + C_{1} = 0 \wedge \log {\left (a \right )} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {\log {\left (- \frac {1}{a^{x} + \log {\left (a^{C_{1}} \right )}} \right )}}{\log {\left (a \right )}} & \text {for}\: \frac {\log {\left (a \right )}}{a^{x} + \log {\left (a^{C_{1}} \right )}} \neq 0 \wedge \log {\left (a \right )} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} + \frac {e^{x \log {\left (a \right )}}}{\log {\left (a \right )}} & \text {for}\: \log {\left (a \right )} = 0 \wedge e^{C_{1} \log {\left (a \right )} + e^{x \log {\left (a \right )}}} \log {\left (a \right )} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} C_{1} + \frac {W\left (x e^{- C_{1} \log {\left (a \right )}} \log {\left (a \right )}\right )}{\log {\left (a \right )}} & \text {for}\: e^{C_{1} \log {\left (a \right )} + W\left (x e^{- C_{1} \log {\left (a \right )}} \log {\left (a \right )}\right )} \log {\left (a \right )} = 0 \wedge \log {\left (a \right )} = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ]
\]