56.20.4 problem Ex 4

Internal problem ID [14199]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter V, Singular solutions. Article 33. Page 73
Problem number : Ex 4
Date solved : Thursday, October 02, 2025 at 09:26:39 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.311 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 60.185 (sec). Leaf size: 3751

ode=y[x]*(3-4*y[x])^2*D[y[x],x]^2==4*(1-y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 2.993 (sec). Leaf size: 228

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

\[ \left [ \begin {cases} 3 i \sqrt {y{\left (x \right )} - 1} \sqrt {y{\left (x \right )}} - \frac {2 i y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} + \frac {3 i \sqrt {y{\left (x \right )}}}{\sqrt {y{\left (x \right )} - 1}} & \text {for}\: \left |{y{\left (x \right )}}\right | > 1 \\\frac {2 y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} - \frac {2 y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} & \text {otherwise} \end {cases} = C_{1} - 2 x, \ \begin {cases} 3 i \sqrt {y{\left (x \right )} - 1} \sqrt {y{\left (x \right )}} - \frac {2 i y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 1}} + \frac {3 i \sqrt {y{\left (x \right )}}}{\sqrt {y{\left (x \right )} - 1}} & \text {for}\: \left |{y{\left (x \right )}}\right | > 1 \\\frac {2 y^{\frac {5}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} - \frac {2 y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {1 - y{\left (x \right )}}} & \text {otherwise} \end {cases} = C_{1} + 2 x\right ] \]