2.252   ODE No. 252

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

\[ \left (x^2 y(x)-1\right ) y'(x)-x y(x)^2+1=0 \] Mathematica : cpu = 10.0071 (sec), leaf count = 819

\[\left \{\left \{y(x)\to \frac {6 x c_1-x}{6 c_1-1}+\frac {\sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}{3 \sqrt [3]{2} (6 c_1-1)}-\frac {\sqrt [3]{2} \left (54 x^2 c_1-9 x^2\right )}{3 (6 c_1-1) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}\right \},\left \{y(x)\to \frac {6 x c_1-x}{6 c_1-1}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}{6 \sqrt [3]{2} (6 c_1-1)}+\frac {\left (1+i \sqrt {3}\right ) \left (54 x^2 c_1-9 x^2\right )}{3\ 2^{2/3} (6 c_1-1) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}\right \},\left \{y(x)\to \frac {6 x c_1-x}{6 c_1-1}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}{6 \sqrt [3]{2} (6 c_1-1)}+\frac {\left (1-i \sqrt {3}\right ) \left (54 x^2 c_1-9 x^2\right )}{3\ 2^{2/3} (6 c_1-1) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}\right \}\right \}\] Maple : cpu = 0.662 (sec), leaf count = 1338

\[\left \{y \left (x \right ) = \frac {\left (c_{1}^{2}-80 c_{1}\right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}+\left (\left (-c_{1}+80\right ) x^{7}-160 x^{4}+80 x \right ) 4^{\frac {1}{3}} \left (c_{1} \left (-\frac {1}{4}+\sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (\left (c_{1}-80\right ) x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}+\left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}}{\left (\left (c_{1}^{2}-80 c_{1}\right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}+\left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) x^{2}+\left (\left (-c_{1}+80\right ) x^{6}-160 x^{3}+80\right ) 4^{\frac {1}{3}} \left (c_{1} \left (-\frac {1}{4}+\sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (\left (c_{1}-80\right ) x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {\left (-c_{1}^{2}+80 c_{1}\right ) x^{8}-160 c_{1} x^{5}+80 c_{1} x^{2}+\left (\left (-2 c_{1}+160\right ) x^{7}-320 x^{4}+160 x \right ) 4^{\frac {1}{3}} \left (c_{1} \left (-\frac {1}{4}+\sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (\left (c_{1}-80\right ) x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}+\left (-i c_{1} \left (c_{1}-80\right ) x^{8}-160 i c_{1} x^{5}+80 i c_{1} x^{2}+i \left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) \sqrt {3}-\left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}}{\left (\left (-c_{1}^{2}+80 c_{1}\right ) x^{8}-160 c_{1} x^{5}+80 c_{1} x^{2}+\left (-i c_{1} \left (c_{1}-80\right ) x^{8}-160 i c_{1} x^{5}+80 i c_{1} x^{2}+i \left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) \sqrt {3}-\left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) x^{2}+\left (\left (-2 c_{1}+160\right ) x^{6}-320 x^{3}+160\right ) 4^{\frac {1}{3}} \left (c_{1} \left (-\frac {1}{4}+\sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (\left (c_{1}-80\right ) x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {\left (c_{1}^{2}-80 c_{1}\right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}+\left (\left (2 c_{1}-160\right ) x^{7}+320 x^{4}-160 x \right ) 4^{\frac {1}{3}} \left (c_{1} \left (-\frac {1}{4}+\sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (\left (c_{1}-80\right ) x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}+\left (-i c_{1} \left (c_{1}-80\right ) x^{8}-160 i c_{1} x^{5}+80 i c_{1} x^{2}+i \left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) \sqrt {3}+\left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}}{\left (\left (c_{1}^{2}-80 c_{1}\right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}+\left (-i c_{1} \left (c_{1}-80\right ) x^{8}-160 i c_{1} x^{5}+80 i c_{1} x^{2}+i \left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) \sqrt {3}+\left (c_{1} \left (-1+4 \sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (c_{1} x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {2}{3}}\right ) x^{2}+\left (\left (2 c_{1}-160\right ) x^{6}+320 x^{3}-160\right ) 4^{\frac {1}{3}} \left (c_{1} \left (-\frac {1}{4}+\sqrt {\frac {-5 x^{6}+10 x^{3}-5}{\left (c_{1}-80\right ) x^{6}+160 x^{3}-80}}\right ) \left (\left (c_{1}-80\right ) x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right \}\]