54.1.450 problem 463

Internal problem ID [11764]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 463
Date solved : Tuesday, September 30, 2025 at 10:19:43 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: 1.722 (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: 15.413 (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)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} \left (C_{1} - e^{x}\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \left (C_{1} - e^{x}\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \left (C_{1} - e^{x}\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} \left (C_{1} + e^{x}\right )^{\frac {2}{3}}}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} + 3 \sqrt [6]{3} i\right ) \left (C_{1} + e^{x}\right )^{\frac {2}{3}}}{4}, \ y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- 3^{\frac {2}{3}} - 3 \sqrt [6]{3} i\right ) \left (C_{1} + e^{x}\right )^{\frac {2}{3}}}{4}\right ] \]