ODE
\[ (2-3 y(x))^2 y'(x)^2=4 (1-y(x)) \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y'\)
Mathematica ✓
cpu = 0.0734112 (sec), leaf count = 891
\[\left \{\left \{y(x)\to \frac {1}{12} \left (2 \sqrt [3]{-108 x^2-108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (2 x+c_1\right ){}^2 \left (108 x^2+108 c_1 x+27 c_1^2-16\right )}+8}+4+\frac {8}{\sqrt [3]{-108 x^2-108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (2 x+c_1\right ){}^2 \left (108 x^2+108 c_1 x+27 c_1^2-16\right )}+8}}\right )\right \},\left \{y(x)\to \frac {1}{24} \left (2 i \left (i+\sqrt {3}\right ) \sqrt [3]{-108 x^2-108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (2 x+c_1\right ){}^2 \left (108 x^2+108 c_1 x+27 c_1^2-16\right )}+8}+8-\frac {8 \left (1+i \sqrt {3}\right )}{\sqrt [3]{-108 x^2-108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (2 x+c_1\right ){}^2 \left (108 x^2+108 c_1 x+27 c_1^2-16\right )}+8}}\right )\right \},\left \{y(x)\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-108 x^2-108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (2 x+c_1\right ){}^2 \left (108 x^2+108 c_1 x+27 c_1^2-16\right )}+8}+8+\frac {8 i \left (i+\sqrt {3}\right )}{\sqrt [3]{-108 x^2-108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (2 x+c_1\right ){}^2 \left (108 x^2+108 c_1 x+27 c_1^2-16\right )}+8}}\right )\right \},\left \{y(x)\to \frac {1}{12} \left (2 \sqrt [3]{-108 x^2+108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (c_1-2 x\right ){}^2 \left (108 x^2-108 c_1 x+27 c_1^2-16\right )}+8}+4+\frac {8}{\sqrt [3]{-108 x^2+108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (c_1-2 x\right ){}^2 \left (108 x^2-108 c_1 x+27 c_1^2-16\right )}+8}}\right )\right \},\left \{y(x)\to \frac {1}{24} \left (2 i \left (i+\sqrt {3}\right ) \sqrt [3]{-108 x^2+108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (c_1-2 x\right ){}^2 \left (108 x^2-108 c_1 x+27 c_1^2-16\right )}+8}+8+\frac {-8-8 i \sqrt {3}}{\sqrt [3]{-108 x^2+108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (c_1-2 x\right ){}^2 \left (108 x^2-108 c_1 x+27 c_1^2-16\right )}+8}}\right )\right \},\left \{y(x)\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-108 x^2+108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (c_1-2 x\right ){}^2 \left (108 x^2-108 c_1 x+27 c_1^2-16\right )}+8}+8+\frac {-8+8 i \sqrt {3}}{\sqrt [3]{-108 x^2+108 c_1 x-27 c_1^2+3 \sqrt {3} \sqrt {\left (c_1-2 x\right ){}^2 \left (108 x^2-108 c_1 x+27 c_1^2-16\right )}+8}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.043 (sec), leaf count = 42
\[ \left \{ x-y \left ( x \right ) \sqrt {1-y \left ( x \right ) }-{\it \_C1}=0,x+y \left ( x \right ) \sqrt {1-y \left ( x \right ) }-{\it \_C1}=0,y \left ( x \right ) =1 \right \} \] Mathematica raw input
DSolve[(2 - 3*y[x])^2*y'[x]^2 == 4*(1 - y[x]),y[x],x]
Mathematica raw output
{{y[x] -> (4 + 8/(8 - 108*x^2 - 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(2*x + C[1])^2*(-16 + 108*x^2 + 108*x*C[1] + 27*C[1]^2)])^(1/3) + 2*(8 - 108*x^2 - 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(2*x + C[1])^2*(-16 + 108*x^2 + 108*x*C[1] + 27*C[1]^2)])^(1/3))/12}, {y[x] -> (8 - (8*(1 + I*Sqrt[3]))/(8 - 108*x^2 - 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(2*x + C[1])^2*(-16 + 108*x^2 + 108*x*C[1] + 27*C[1]^2)])^(1/3) + (2*I)*(I + Sqrt[3])*(8 - 108*x^2 - 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(2*x + C[1])^2*(-16 + 108*x^2 + 108*x*C[1] + 27*C[1]^2)])^(1/3))/24}, {y[x] -> (8 + ((8*I)*(I + Sqrt[3]))/(8 - 108*x^2 - 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(2*x + C[1])^2*(-16 + 108*x^2 + 108*x*C[1] + 27*C[1]^2)])^(1/3) - 2*(1 + I*Sqrt[3])*(8 - 108*x^2 - 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(2*x + C[1])^2*(-16 + 108*x^2 + 108*x*C[1] + 27*C[1]^2)])^(1/3))/24}, {y[x] -> (4 + 8/(8 - 108*x^2 + 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(-2*x + C[1])^2*(-16 + 108*x^2 - 108*x*C[1] + 27*C[1]^2)])^(1/3) + 2*(8 - 108*x^2 + 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(-2*x + C[1])^2*(-16 + 108*x^2 - 108*x*C[1] + 27*C[1]^2)])^(1/3))/12}, {y[x] -> (8 + (-8 - (8*I)*Sqrt[3])/(8 - 108*x^2 + 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(-2*x + C[1])^2*(-16 + 108*x^2 - 108*x*C[1] + 27*C[1]^2)])^(1/3) + (2*I)*(I + Sqrt[3])*(8 - 108*x^2 + 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(-2*x + C[1])^2*(-16 + 108*x^2 - 108*x*C[1] + 27*C[1]^2)])^(1/3))/24}, {y[x] -> (8 + (-8 + (8*I)*Sqrt[3])/(8 - 108*x^2 + 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(-2*x + C[1])^2*(-16 + 108*x^2 - 108*x*C[1] + 27*C[1]^2)])^(1/3) - 2*(1 + I*Sqrt[3])*(8 - 108*x^2 + 108*x*C[1] - 27*C[1]^2 + 3*Sqrt[3]*Sqrt[(-2*x + C[1])^2*(-16 + 108*x^2 - 108*x*C[1] + 27*C[1]^2)])^(1/3))/24}}
Maple raw input
dsolve((2-3*y(x))^2*diff(y(x),x)^2 = 4-4*y(x), y(x),'implicit')
Maple raw output
y(x) = 1, x-y(x)*(1-y(x))^(1/2)-_C1 = 0, x+y(x)*(1-y(x))^(1/2)-_C1 = 0