60.1.450 problem 463
Internal
problem
ID
[10464]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
463
Date
solved
:
Sunday, March 30, 2025 at 04:50:41 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} y {y^{\prime }}^{2}-{\mathrm e}^{2 x}&=0 \end{align*}
✓ Maple. Time used: 0.044 (sec). Leaf size: 67
ode:=y(x)*diff(y(x),x)^2-exp(2*x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {2 y^{2}+3 c_1 \sqrt {y}-3 \sqrt {y \,{\mathrm e}^{2 x}}}{3 \sqrt {y}} &= 0 \\
\frac {2 y^{2}+3 c_1 \sqrt {y}+3 \sqrt {y \,{\mathrm e}^{2 x}}}{3 \sqrt {y}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 2.111 (sec). Leaf size: 47
ode=-E^(2*x) + y[x]*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (-e^x+c_1\right ){}^{2/3} \\
y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (e^x+c_1\right ){}^{2/3} \\
\end{align*}
✓ Sympy. Time used: 21.064 (sec). Leaf size: 170
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(y(x)*Derivative(y(x), x)**2 - exp(2*x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} \left (C_{1} - e^{x}\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \left (C_{1} - e^{x}\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \left (C_{1} - e^{x}\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} \left (C_{1} + e^{x}\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \left (C_{1} + e^{x}\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \left (C_{1} + e^{x}\right )^{\frac {2}{3}}}{4}\right ]
\]