29.34.1 problem 996

Internal problem ID [5572]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 996
Date solved : Monday, January 27, 2025 at 12:07:16 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 469

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

\begin{align*} y \left (x \right ) &= 1 \\ y \left (x \right ) &= -\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 \left (x \right ) &= 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 \left (x \right ) &= 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 \left (x \right ) &= -\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 \left (x \right ) &= 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 \left (x \right ) &= 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*}

✓ Solution by Mathematica

Time used: 4.393 (sec). Leaf size: 896

DSolve[(2-3 y[x])^2 (D[y[x],x])^2 ==4(1-y[x]),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*}