23.2.243 problem 249

Internal problem ID [5598]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 249
Date solved : Tuesday, September 30, 2025 at 01:10:51 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 469

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

\begin{align*} y &= 1 \\ y &= -\frac {{\left (\left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{1}/{3}}+\frac {12}{\left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{1}/{3}}}\right )}^{2}}{36}+1 \\ y &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\ y &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\ y &= -\frac {{\left (\left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{1}/{3}}+\frac {12}{\left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{1}/{3}}}\right )}^{2}}{36}+1 \\ y &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\ y &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 3.533 (sec). Leaf size: 896

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

\begin{align*} y(x)&\to \frac {1}{12} \left (2 \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}+\frac {8}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}}+4\right )\\ y(x)&\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}-\frac {8 \left (1+i \sqrt {3}\right )}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}}+8\right )\\ y(x)&\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}+\frac {8 i \left (\sqrt {3}+i\right )}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}}+8\right )\\ y(x)&\to \frac {1}{12} \left (2 \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}+\frac {8}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}}+4\right )\\ y(x)&\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}+\frac {-8-8 i \sqrt {3}}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}}+8\right )\\ y(x)&\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}+\frac {-8+8 i \sqrt {3}}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}}+8\right )\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.614 (sec). Leaf size: 214

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

\[ \left [ \begin {cases} - \frac {2 \sqrt {3} i \left (y{\left (x \right )} - \frac {2}{3}\right ) \sqrt {3 y{\left (x \right )} - 3}}{3} - \frac {4 \sqrt {3} i \sqrt {3 y{\left (x \right )} - 3}}{9} & \text {for}\: \left |{y{\left (x \right )} - \frac {2}{3}}\right | > \frac {1}{3} \\- \frac {2 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}} \left (y{\left (x \right )} - \frac {2}{3}\right )}{3} - \frac {4 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}}}{9} & \text {otherwise} \end {cases} = C_{1} - 2 x, \ \begin {cases} - \frac {2 \sqrt {3} i \left (y{\left (x \right )} - \frac {2}{3}\right ) \sqrt {3 y{\left (x \right )} - 3}}{3} - \frac {4 \sqrt {3} i \sqrt {3 y{\left (x \right )} - 3}}{9} & \text {for}\: \left |{y{\left (x \right )} - \frac {2}{3}}\right | > \frac {1}{3} \\- \frac {2 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}} \left (y{\left (x \right )} - \frac {2}{3}\right )}{3} - \frac {4 \sqrt {3} \sqrt {3 - 3 y{\left (x \right )}}}{9} & \text {otherwise} \end {cases} = C_{1} + 2 x\right ] \]