77.1.54 problem 73 (page 112)

Internal problem ID [17865]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 73 (page 112)
Date solved : Thursday, March 13, 2025 at 11:04:42 AM
CAS classification : [_quadrature]

\begin{align*} y^{2} \left (y^{\prime }-1\right )&=\left (2-y^{\prime }\right )^{2} \end{align*}

✓ Maple. Time used: 0.163 (sec). Leaf size: 75

ode:=y(x)^2*(diff(y(x),x)-1) = (2-diff(y(x),x))^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -2 i \\ y &= 2 i \\ x +2 \left (\int _{}^{y}\frac {1}{-\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2} \left (\textit {\_a}^{2}+4\right )}-4}d \textit {\_a} \right )-c_{1} &= 0 \\ x -2 \left (\int _{}^{y}\frac {1}{\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2} \left (\textit {\_a}^{2}+4\right )}+4}d \textit {\_a} \right )-c_{1} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.539 (sec). Leaf size: 73

ode=y[x]^2*(D[y[x],x]-1)==(2-D[y[x],x])^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {x^2-4 c_1 x-1+4 c_1{}^2}{x-2 c_1} \\ y(x)\to \frac {x^2+4 c_1 x-1+4 c_1{}^2}{x+2 c_1} \\ y(x)\to -2 i \\ y(x)\to 2 i \\ \end{align*}

✓ Sympy. Time used: 3.711 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(2 - Derivative(y(x), x))**2 + (Derivative(y(x), x) - 1)*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ - 2 \int \limits ^{y{\left (x \right )}} \frac {1}{y^{2} - y \sqrt {y^{2} + 4} + 4}\, dy = C_{1} - x, \ - 2 \int \limits ^{y{\left (x \right )}} \frac {1}{y^{2} + y \sqrt {y^{2} + 4} + 4}\, dy = C_{1} - x\right ] \]