54.1.362 problem 371

Internal problem ID [11676]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 371
Date solved : Tuesday, September 30, 2025 at 10:06:14 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-y^{3}+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 22

ode:=diff(y(x),x)^2-y(x)^3+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 1 \\ y &= 0 \\ y &= \sec \left (-\frac {x}{2}+\frac {c_1}{2}\right )^{2} \\ \end{align*}

✓ Mathematica. Time used: 0.667 (sec). Leaf size: 45

ode=y[x]^2 - y[x]^3 + D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sec ^2\left (\frac {x-c_1}{2}\right )\\ y(x)&\to 1+\tan ^2\left (\frac {x+c_1}{2}\right )\\ y(x)&\to 0\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.450 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**3 + y(x)**2 + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \begin {cases} 2 i \operatorname {acosh}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {for}\: \frac {1}{\left |{y{\left (x \right )}}\right |} > 1 \\- 2 \operatorname {asin}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {otherwise} \end {cases} = C_{1} - x, \ \begin {cases} 2 i \operatorname {acosh}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {for}\: \frac {1}{\left |{y{\left (x \right )}}\right |} > 1 \\- 2 \operatorname {asin}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {otherwise} \end {cases} = C_{1} + x\right ] \]