54.1.485 problem 498

Internal problem ID [11799]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 498
Date solved : Tuesday, September 30, 2025 at 10:43:48 PM
CAS classification : [_quadrature]

\begin{align*} \left (3 y-2\right ) {y^{\prime }}^{2}-4+4 y&=0 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 99

ode:=(3*y(x)-2)*diff(y(x),x)^2-4+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 1 \\ y &= \frac {\sin \left (\operatorname {RootOf}\left (8 \sqrt {3}\, c_1 \textit {\_Z} -8 \sqrt {3}\, x \textit {\_Z} +\cos \left (\textit {\_Z} \right )^{2}-48 c_1^{2}+96 c_1 x -48 x^{2}-\textit {\_Z}^{2}\right )\right )}{6}+\frac {5}{6} \\ y &= \frac {\sin \left (\operatorname {RootOf}\left (8 \sqrt {3}\, c_1 \textit {\_Z} -8 \sqrt {3}\, x \textit {\_Z} -\cos \left (\textit {\_Z} \right )^{2}+48 c_1^{2}-96 c_1 x +48 x^{2}+\textit {\_Z}^{2}\right )\right )}{6}+\frac {5}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.315 (sec). Leaf size: 160

ode=-4 + 4*y[x] + (-2 + 3*y[x])*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}} \text {arcsinh}\left (\sqrt {3} \sqrt {\text {$\#$1}-1}\right )}{\sqrt {3} \sqrt {\text {$\#$1}-1}}-\sqrt {3 (\text {$\#$1}-1)+1} \sqrt {1-\text {$\#$1}}\&\right ][-2 x+c_1]\\ y(x)&\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}} \text {arcsinh}\left (\sqrt {3} \sqrt {\text {$\#$1}-1}\right )}{\sqrt {3} \sqrt {\text {$\#$1}-1}}-\sqrt {3 (\text {$\#$1}-1)+1} \sqrt {1-\text {$\#$1}}\&\right ][2 x+c_1]\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 1.436 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*y(x) - 2)*Derivative(y(x), x)**2 + 4*y(x) - 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {1 - y}{3 y - 2}}}\, dy = C_{1} + 2 x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {1 - y}{3 y - 2}}}\, dy = C_{1} - 2 x\right ] \]