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 = 0.254756 (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 {(2 x+c_1){}^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 {(2 x+c_1){}^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 {(2 x+c_1){}^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 {(2 x+c_1){}^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 {(2 x+c_1){}^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 {(2 x+c_1){}^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 {(c_1-2 x){}^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 {(c_1-2 x){}^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 {(c_1-2 x){}^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 {(c_1-2 x){}^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 {(c_1-2 x){}^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 {(c_1-2 x){}^2 \left (108 x^2-108 c_1 x+27 c_1{}^2-16\right )}+8}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.203 (sec), leaf count = 713
\[\left [y \left (x \right ) = 1, y \left (x \right ) = -\left (\frac {\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )^{2}+1, y \left (x \right ) = -\left (-\frac {\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1, y \left (x \right ) = -\left (-\frac {\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (-108 x +108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1, y \left (x \right ) = -\left (\frac {\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )^{2}+1, y \left (x \right ) = -\left (-\frac {\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1, y \left (x \right ) = -\left (-\frac {\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 x -108 \textit {\_C1} +12 \sqrt {81 \textit {\_C1}^{2}-162 x \textit {\_C1} +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+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))
Maple raw output
[y(x) = 1, y(x) = -(1/6*(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)+2/(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3))^2+1, y(x) = -(-1/12*(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-1/(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-2/(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)))^2+1, y(x) = -(-1/12*(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-1/(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/6*(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-2/(-108*x+108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)))^2+1, y(x) = -(1/6*(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)+2/(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3))^2+1, y(x) = -(-1/12*(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-1/(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-2/(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)))^2+1, y(x) = -(-1/12*(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-1/(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/6*(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)-2/(108*x-108*_C1+12*(81*_C1^2-162*_C1*x+81*x^2-12)^(1/2))^(1/3)))^2+1]