2.17.20 Problems 1901 to 2000

Problem 1901

ODE	\[ \boxed {\left (x +y\right ) y^{\prime }-y=-x} \]
program solution	\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}-\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right ) x \]

Problem 1902

ODE	\[ \boxed {-y+y^{\prime } x -\sqrt {y x}=0} \]
program solution	\[ y = \frac {x \ln \left (x \right )^{2}}{4}-\frac {c_{1} x \ln \left (x \right )}{2}+\frac {c_{1}^{2} x}{4} \] Verified OK.
Maple solution	\[ -\frac {y \left (x \right )}{\sqrt {x y \left (x \right )}}+\frac {\ln \left (x \right )}{2}-c_{1} = 0 \]

Problem 1903

ODE	\[ \boxed {y^{\prime }-\frac {2 x -y}{x +4 y}=0} \]
program solution	\[ -x \left (x -y\right )+2 y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-c_{1} x -\sqrt {9 c_{1}^{2} x^{2}+8}}{4 c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {9 c_{1}^{2} x^{2}+8}}{4 c_{1}} \\ \end{align*}

Problem 1904

ODE	\[ \boxed {-y+y^{\prime } x -\sqrt {x^{2}-y^{2}}=0} \]
program solution	\[ y = -\tan \left (-\ln \left (x \right )+c_{1} \right ) \sqrt {\frac {x^{2}}{\tan \left (-\ln \left (x \right )+c_{1} \right )^{2}+1}} \] Verified OK.
Maple solution	\[ -\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )+\ln \left (x \right )-c_{1} = 0 \]

Problem 1905

ODE	\[ \boxed {y y^{\prime }-2 y=-x} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (x \,{\mathrm e}^{-c_{1}}\right )+c_{1}}+x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \left (\operatorname {LambertW}\left (c_{1} x \right )+1\right )}{\operatorname {LambertW}\left (c_{1} x \right )} \]

Problem 1906

ODE	\[ \boxed {y^{\prime } x -y+\sqrt {-x^{2}+y^{2}}=0} \]
program solution	\[ y = \frac {\left ({\mathrm e}^{2 c_{1}}+x^{2}\right ) {\mathrm e}^{-c_{1}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right )+\sqrt {y \left (x \right )^{2}-x^{2}}-c_{1} = 0 \]

Problem 1907

ODE	\[ \boxed {y^{2}-y^{\prime } x y=-x^{2}} \]
program solution	\[ -\frac {y^{2}}{2 x^{2}}+\ln \left (x \right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {2 \ln \left (x \right )+c_{1}}\, x \\ y \left (x \right ) &= -\sqrt {2 \ln \left (x \right )+c_{1}}\, x \\ \end{align*}

Problem 1908

ODE	\[ \boxed {\left (-x^{2}+y x \right ) y^{\prime }-y^{2}=0} \]
program solution	\[ y = {\mathrm e}^{-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-c_{1}}}{x}\right )-c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -x \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-c_{1}}}{x}\right ) \]

Problem 1909

ODE	\[ \boxed {y+y^{\prime } x -2 \sqrt {y x}=0} \]
program solution	\[ y = \frac {c_{1}^{2}+4 c_{1} x +4 x^{2}}{4 x} \] Verified OK.
Maple solution	\[ \frac {y \left (x \right ) c_{1} x^{2}-\sqrt {x y \left (x \right )}\, y \left (x \right ) c_{1} x -c_{1} x^{3}+\sqrt {x y \left (x \right )}\, c_{1} x^{2}+x +\sqrt {x y \left (x \right )}}{\left (-x +y \left (x \right )\right ) \left (\sqrt {x y \left (x \right )}-x \right ) x} = 0 \]

Problem 1910

ODE	\[ \boxed {y+\left (x -y\right ) y^{\prime }=-x} \]
program solution	\[ \frac {x \left (x +2 y\right )}{2}-\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {c_{1} x -\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {c_{1} x +\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*}

Problem 1911

ODE	\[ \boxed {y \left (y^{2}-y x +x^{2}\right )+x y^{\prime } \left (x^{2}+y x +y^{2}\right )=0} \]
program solution	\[ \frac {\arctan \left (\frac {y}{x}\right )}{2}+\frac {\ln \left (y\right )}{2} = -\frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (\ln \left (\tan \left (\textit {\_Z} \right )\right )+\textit {\_Z} +2 \ln \left (x \right )+2 c_{1} \right )\right ) x \]

Problem 1912

ODE	\[ \boxed {y^{\prime } x -y-x \sin \left (\frac {y}{x}\right )=0} \]
program solution	\[ y = \arctan \left (-\frac {2 c_{1} x}{c_{1}^{2} x^{2}+1}, -\frac {c_{1}^{2} x^{2}-1}{c_{1}^{2} x^{2}+1}\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arctan \left (\frac {2 x c_{1}}{c_{1}^{2} x^{2}+1}, \frac {-c_{1}^{2} x^{2}+1}{c_{1}^{2} x^{2}+1}\right ) x \]

Problem 1913

ODE	\[ \boxed {y^{\prime }-\frac {y}{x}-\cosh \left (\frac {y}{x}\right )=0} \]
program solution	\[ y = -\operatorname {arcsinh}\left (\tan \left (\ln \left (-\frac {1}{c_{1} x}\right )\right )\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (\tan \left (\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )\right ) x \]

Problem 1914

ODE	\[ \boxed {y^{2}-2 y^{\prime } x y=-x^{2}} \] With initial conditions \begin {align*} [y \left (-1\right ) = 0] \end {align*}
program solution	\[ x -\frac {y^{2}}{x} = -1 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {\left (x +1\right ) x} \\ y \left (x \right ) &= -\sqrt {\left (x +1\right ) x} \\ \end{align*}

Problem 1915

ODE	\[ \boxed {\left (\frac {x}{y}+\frac {y}{x}\right ) y^{\prime }=-1} \]
program solution	\[ \frac {x^{2} y^{2}}{2}+\frac {y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= -\frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y \left (x \right ) &= \frac {\sqrt {x^{2} c_{1} \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right )}}{x \left (-c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ \end{align*}

Problem 1916

ODE	\[ \boxed {{\mathrm e}^{\frac {y}{x}} x +y-y^{\prime } x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = -\ln \left (1+\ln \left (\frac {1}{x}\right )\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (-\frac {1}{-1+\ln \left (x \right )}\right ) x \]

Problem 1917

ODE	\[ \boxed {y^{\prime }-\frac {x +y}{x -y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} -\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )-2 \ln \left (x \right )\right )\right ) x \]

Problem 1918

ODE	\[ \boxed {y^{\prime }-\frac {y}{x}-\tan \left (\frac {y}{x}\right )=0} \] With initial conditions \begin {align*} [y \left (6\right ) = \pi ] \end {align*}
program solution	\[ y = \arcsin \left (\frac {x}{12}\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arcsin \left (\frac {x}{12}\right ) x \]

Problem 1919

ODE	\[ \boxed {\left (3 y x -2 x^{2}\right ) y^{\prime }-2 y^{2}+y x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
program solution	\[ -\frac {y^{2} \left (x -y\right )}{x^{2}} = -2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-27 x^{2}+x^{3}+3 \sqrt {3}\, \sqrt {-x^{4} \left (2 x -27\right )}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{6}-\frac {\left (i \sqrt {3}\, x +x -2 \left (-27 x^{2}+x^{3}+3 \sqrt {3}\, \sqrt {-x^{4} \left (2 x -27\right )}\right )^{\frac {1}{3}}\right ) x}{6 \left (-27 x^{2}+x^{3}+3 \sqrt {3}\, \sqrt {-x^{4} \left (2 x -27\right )}\right )^{\frac {1}{3}}} \]

Problem 1920

ODE	\[ \boxed {y^{\prime }-\frac {y}{x -k \sqrt {x^{2}+y^{2}}}=0} \]
program solution	\[ \frac {\ln \left (x +\sqrt {x^{2}+y^{2}}\right )+\left (k -1\right ) \ln \left (y\right )-c_{1} k +\ln \left (2\right )}{k} = 0 \] Verified OK.
Maple solution	\[ -c_{1} +y \left (x \right )^{k -1} \sqrt {x^{2}+y \left (x \right )^{2}}+x y \left (x \right )^{k -1} = 0 \]

Problem 1921

ODE	\[ \boxed {y^{2} \left (y y^{\prime }-x \right )=-x^{3}} \]
program solution	\[ \frac {\ln \left (y^{4}-x^{2} y^{2}+x^{4}\right )}{4}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 y^{2}+x^{2}\right ) \sqrt {3}}{3 x^{2}}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (2 \textit {\_Z}^{2}+\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{4}\right )+\sqrt {3}\, \ln \left (3\right )-2 \sqrt {3}\, \ln \left (2\right )+4 \sqrt {3}\, c_{1} -2 \textit {\_Z} \right )\right )-1\right ) x \]

Problem 1922

ODE	\[ \boxed {y^{\prime }-\frac {y}{x}-\tanh \left (\frac {y}{x}\right )=0} \]
program solution	\[ y = -\operatorname {arcsinh}\left (c_{1} x \right ) x \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \operatorname {arctanh}\left (\frac {-c_{1} x^{2}+\sqrt {c_{1} x^{2} \left (c_{1} x^{2}-1\right )}}{-c_{1} x^{2}+\sqrt {c_{1} x^{2} \left (c_{1} x^{2}-1\right )}+1}\right ) x \\ y \left (x \right ) &= \operatorname {arctanh}\left (\frac {c_{1} x^{2}+\sqrt {c_{1} x^{2} \left (c_{1} x^{2}-1\right )}}{c_{1} x^{2}-1+\sqrt {c_{1} x^{2} \left (c_{1} x^{2}-1\right )}}\right ) x \\ \end{align*}

Problem 1923

ODE	\[ \boxed {y-\left (x -y+2\right ) y^{\prime }=-x} \]
program solution	\[ -\frac {\ln \left (y^{2}+x^{2}-2 y+2 x +2\right )}{2}+\arctan \left (\frac {y-1}{x +1}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x +1\right )+2 c_{1} \right )\right ) \left (-x -1\right ) \]

Problem 1924

ODE	\[ \boxed {\left (x -2 y+2\right ) y^{\prime }=-x} \]
program solution	\[ \frac {2 \ln \left (x +2 y-2\right )}{3}+\frac {\ln \left (-1-x +y\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {-2 \left (2 c_{1} x^{3}+2 \sqrt {-2 \left (c_{1} x^{3}-\frac {1}{2}\right ) c_{1}^{2} x^{6}}\right )^{\frac {2}{3}} c_{1} x^{2}-\frac {\left (c_{1} x^{3}+\sqrt {-2 \left (c_{1} x^{3}-\frac {1}{2}\right ) c_{1}^{2} x^{6}}\right ) \left (1+i \sqrt {3}\right ) \left (2 c_{1} x^{3}+2 \sqrt {-2 \left (c_{1} x^{3}-\frac {1}{2}\right ) c_{1}^{2} x^{6}}\right )^{\frac {1}{3}}}{2}+c_{1}^{2} \left (i \sqrt {3}-1\right ) x^{6}}{2 \left (2 c_{1} x^{3}+2 \sqrt {-2 \left (c_{1} x^{3}-\frac {1}{2}\right ) c_{1}^{2} x^{6}}\right )^{\frac {2}{3}} x^{2} c_{1}} \]

Problem 1925

ODE	\[ \boxed {-y+\left (x +y\right ) y^{\prime }=-1-2 x} \]
program solution	\[ -\frac {\ln \left (3 y^{2}+6 x^{2}-2 y+4 x +1\right )}{6}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (3 y-1\right ) \sqrt {2}}{6 x +2}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{3}+\frac {\tan \left (\operatorname {RootOf}\left (\sqrt {2}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} \left (3 x +1\right )^{2}\right )+\sqrt {2}\, \ln \left (2\right )+2 \sqrt {2}\, c_{1} -2 \textit {\_Z} \right )\right ) \sqrt {2}\, \left (-3 x -1\right )}{3} \]

Problem 1926

ODE	\[ \boxed {-y+\left (x +y-1\right ) y^{\prime }=-x -2} \]
program solution	\[ \frac {\ln \left (2 y^{2}+2 x^{2}-6 y+2 x +5\right )}{4}+\frac {\arctan \left (\frac {2 y-3}{1+2 x}\right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3}{2}+\tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (2 x +1\right )+2 c_{1} \right )\right ) \left (-\frac {1}{2}-x \right ) \]

Problem 1927

ODE	\[ \boxed {-y+\left (1-x +y\right ) y^{\prime }=-x} \]
program solution	\[ \frac {x \left (x -2 y\right )}{2}+\frac {y^{2}}{2}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x -1-\sqrt {2 c_{1} -2 x +1} \\ y \left (x \right ) &= x -1+\sqrt {2 c_{1} -2 x +1} \\ \end{align*}

Problem 1928

ODE	\[ \boxed {y^{\prime }-\frac {x +y-1}{x -y-1}=0} \]
program solution	\[ -\frac {\ln \left (y^{2}+x^{2}-2 x +1\right )}{2}+\arctan \left (\frac {y}{x -1}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -1\right )+2 c_{1} \right )\right ) \left (1-x \right ) \]

Problem 1929

ODE	\[ \boxed {y+\left (2 x +2 y-1\right ) y^{\prime }=-x} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left (2 \,{\mathrm e}^{x -2+c_{1}}\right )}{2}-x +1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {LambertW}\left (2 \,{\mathrm e}^{x -2-c_{1}}\right )}{2}-x +1 \]

Problem 1930

ODE	\[ \boxed {-y+\left (x -y-1\right ) y^{\prime }=-x -1} \]
program solution	\[ y = x +\operatorname {LambertW}\left ({\mathrm e}^{-2 x +2 c_{1}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {LambertW}\left ({\mathrm e}^{-2 x} c_{1} \right )+x \]

Problem 1931

ODE	\[ \boxed {2 y+\left (3 x +6 y+3\right ) y^{\prime }=-x} \]
program solution	\[ y = -\frac {x}{2}-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{6}-\frac {c_{1}}{3}-\frac {3}{2}}}{2}\right )-\frac {3}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{6}-\frac {3}{2}+\frac {c_{1}}{6}}}{2}\right )-\frac {3}{2}-\frac {x}{2} \]

Problem 1932

ODE	\[ \boxed {2 y-\left (2 x +y-1\right ) y^{\prime }=-x -2} \]
program solution	\[ \frac {\ln \left (3 x +3 y+1\right )}{4}-\frac {3 \ln \left (y+3-x \right )}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x -3\right ) \left (i \sqrt {3}-1\right ) \left (486 \sqrt {\left (x -\frac {4}{3}\right )^{2} c_{1} \left (-\frac {1}{243}+\left (x -\frac {4}{3}\right )^{2} c_{1} \right )}+1-486 \left (x -\frac {4}{3}\right )^{2} c_{1} \right )^{\frac {2}{3}}+\left (-10 x +10\right ) \left (486 \sqrt {\left (x -\frac {4}{3}\right )^{2} c_{1} \left (-\frac {1}{243}+\left (x -\frac {4}{3}\right )^{2} c_{1} \right )}+1-486 \left (x -\frac {4}{3}\right )^{2} c_{1} \right )^{\frac {1}{3}}-\left (x -3\right ) \left (1+i \sqrt {3}\right )}{i \left (486 \sqrt {\left (x -\frac {4}{3}\right )^{2} c_{1} \left (-\frac {1}{243}+\left (x -\frac {4}{3}\right )^{2} c_{1} \right )}+1-486 \left (x -\frac {4}{3}\right )^{2} c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}-\left (486 \sqrt {\left (x -\frac {4}{3}\right )^{2} c_{1} \left (-\frac {1}{243}+\left (x -\frac {4}{3}\right )^{2} c_{1} \right )}+1-486 \left (x -\frac {4}{3}\right )^{2} c_{1} \right )^{\frac {2}{3}}+2 \left (486 \sqrt {\left (x -\frac {4}{3}\right )^{2} c_{1} \left (-\frac {1}{243}+\left (x -\frac {4}{3}\right )^{2} c_{1} \right )}+1-486 \left (x -\frac {4}{3}\right )^{2} c_{1} \right )^{\frac {1}{3}}-1} \]

Problem 1933

ODE	\[ \boxed {-y+\left (x -3 y-5\right ) y^{\prime }=-3 x -1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ 2 \ln \left (3+y+x \right )+\ln \left (1-x +y\right ) = 2 \ln \left (3\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-324+12 \sqrt {96 x^{3}+288 x^{2}+288 x +825}\right )^{\frac {4}{3}}-12 \left (-324+12 \sqrt {96 x^{3}+288 x^{2}+288 x +825}\right )^{\frac {2}{3}} x -84 \left (-324+12 \sqrt {96 x^{3}+288 x^{2}+288 x +825}\right )^{\frac {2}{3}}+576 x^{2}+1152 x +576}{36 \left (-324+12 \sqrt {96 x^{3}+288 x^{2}+288 x +825}\right )^{\frac {2}{3}}} \]

Problem 1934

ODE	\[ \boxed {-3 y+\left (2 x -y+5\right ) y^{\prime }=-6 x -6} \] With initial conditions \begin {align*} [y \left (-1\right ) = 1] \end {align*}
program solution	\[ y = -\frac {9 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {25 x}{9}+\frac {26}{9}}}{9}\right )}{5}+2 x +\frac {16}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {16}{5}-\frac {9 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {25 x}{9}+\frac {26}{9}}}{9}\right )}{5}+2 x \]

Problem 1935

ODE	\[ \boxed {3 y+\left (-x +y\right ) y^{\prime }=-2 x -2} \] With initial conditions \begin {align*} [y \left (0\right ) = -2] \end {align*}
program solution	\[ \frac {\ln \left (10 x^{2}+\left (10 y+12\right ) x +5 y^{2}+8 y+4\right )}{5}-\frac {4 \arctan \left (\frac {5 y+4+5 x}{5 x +2}\right )}{5} = \frac {3 \ln \left (2\right )}{5}+\frac {4 \arctan \left (3\right )}{5} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 1936

ODE	\[ \boxed {y-\left (2 x +2 y-1\right ) y^{\prime }=-x -4} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {3 \operatorname {LambertW}\left (-\frac {2 \,{\mathrm e}^{-x -\frac {2}{3}}}{3}\right )}{2}-x -1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -x -\frac {3 \operatorname {LambertW}\left (-\frac {2 \,{\mathrm e}^{-x -\frac {2}{3}}}{3}\right )}{2}-1 \]

Problem 1937

ODE	\[ \boxed {3 y+\left (2 x +3 y+2\right ) y^{\prime }=1-2 x} \] With initial conditions \begin {align*} [y \left (3\right ) = 1] \end {align*}
program solution	\[ y = 3 \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{-\frac {x}{9}+\frac {5}{9}}}{9}\right )-\frac {2 x}{3}+\frac {7}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {7}{3}-\frac {2 x}{3}+3 \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {5}{9}-\frac {x}{9}}}{9}\right ) \]

Problem 1938

ODE	\[ \boxed {-y+\left (x +2 y+1\right ) y^{\prime }=-3 x -2} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ \frac {\ln \left (21 x^{2}+14 y^{2}+30 x +4 y+11\right )}{7}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (7 y+1\right ) \sqrt {6}}{21 x +15}\right )}{21} = \frac {\ln \left (11\right )}{7}+\frac {\sqrt {6}\, \arctan \left (\frac {\sqrt {6}}{15}\right )}{21} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 1939

ODE	\[ \boxed {2 y-\left (x +2 y-1\right ) y^{\prime }=-3 x -3} \] With initial conditions \begin {align*} [y \left (-2\right ) = 1] \end {align*}
program solution	\[ \frac {4 \ln \left (2 y-3 x -9\right )}{5}+\frac {\ln \left (2 x +2 y+1\right )}{5} = i \pi \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2-x \right ) {\operatorname {RootOf}\left (-1+\left (x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x +32\right ) \textit {\_Z}^{25}+\left (-5 x^{5}-50 x^{4}-200 x^{3}-400 x^{2}-400 x -160\right ) \textit {\_Z}^{20}\right )}^{5}}{2}+\frac {3 x}{2}+\frac {9}{2} \]

Problem 1940

ODE	\[ \boxed {-2 y+\left (1-x +2 y\right ) y^{\prime }=-x -3} \] With initial conditions \begin {align*} [y \left (-4\right ) = 2] \end {align*}
program solution	\[ y = 4 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {x}{8}+\frac {5}{8}}}{8}\right )+\frac {x}{2}+\frac {7}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {7}{2}+\frac {x}{2}+4 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {x}{8}+\frac {5}{8}}}{8}\right ) \]

Problem 1941

ODE	\[ \boxed {y+\left (4 x +2 y+1\right ) y^{\prime }=-2 x} \] With initial conditions \begin {align*} \left [y \left (-\frac {1}{6}\right ) = 0\right ] \end {align*}
program solution	\[ y = -\frac {\operatorname {LambertW}\left (-1, -2 \,{\mathrm e}^{-9 x -\frac {7}{2}}\right )}{6}-2 x -\frac {2}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\operatorname {LambertW}\left (-1, -2 \,{\mathrm e}^{-9 x -\frac {7}{2}}\right )}{6}-\frac {2}{3}-2 x \]

Problem 1942

ODE	\[ \boxed {y+\left (4 x -2 y+1\right ) y^{\prime }=-2 x} \] With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = 0\right ] \end {align*}
program solution	\[ \frac {\ln \left (-16 x^{2}+\left (-40 y+6\right ) x +16 y^{2}-13 y+2\right )}{8}-\frac {3 \sqrt {41}\, \operatorname {arctanh}\left (\frac {\left (-32 y+13+40 x \right ) \sqrt {41}}{328 x +41}\right )}{164} = \frac {3 \sqrt {41}\, \left (i \pi -2 \,\operatorname {arccoth}\left (\frac {33 \sqrt {41}}{205}\right )\right )}{328} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (6 \sqrt {41}\, \operatorname {arctanh}\left (\frac {\left (-13+32 \textit {\_Z} -40 x \right ) \sqrt {41}}{328 x +41}\right )+41 \ln \left (\frac {16 \textit {\_Z}^{2}-40 x \textit {\_Z} -16 x^{2}-13 \textit {\_Z} +6 x +2}{\left (8 x +1\right )^{2}}\right )+82 \ln \left (8 x +1\right )+6 \sqrt {41}\, \operatorname {arctanh}\left (\frac {33 \sqrt {41}}{205}\right )\right ) \]

Problem 1943

ODE	\[ \boxed {y+\left (x -2 y\right ) y^{\prime }=-x} \]
program solution	\[ \frac {x \left (x +2 y\right )}{2}-y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {c_{1} x -\sqrt {3 c_{1}^{2} x^{2}+2}}{2 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} x +\sqrt {3 c_{1}^{2} x^{2}+2}}{2 c_{1}} \\ \end{align*}

Problem 1944

ODE	\[ \boxed {y+\left (x +3 y\right ) y^{\prime }=-3 x} \]
program solution	\[ \frac {x \left (3 x +2 y\right )}{2}+\frac {3 y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-c_{1} x -\sqrt {-8 c_{1}^{2} x^{2}+3}}{3 c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {-8 c_{1}^{2} x^{2}+3}}{3 c_{1}} \\ \end{align*}

Problem 1945

ODE	\[ \boxed {b_{1} y+\left (b_{1} x +b_{2} y+c_{2} \right ) y^{\prime }=-a_{1} x -c_{1}} \]
program solution	\[ \frac {x \left (a_{1} x +2 b_{1} y+2 c_{1} \right )}{2}+\frac {b_{2} y^{2}}{2}+c_{2} y = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {-\left (a_{1} b_{2} -b_{1}^{2}\right ) \left (\left (a_{1} x +c_{1} \right ) b_{2} -b_{1}^{2} x -c_{2} b_{1} \right )^{2} {\mathrm e}^{2 c_{1}}+b_{2}}\, {\mathrm e}^{-c_{1}}-\left (a_{1} b_{2} -b_{1}^{2}\right ) \left (b_{1} x +c_{2} \right )}{\left (a_{1} b_{2} -b_{1}^{2}\right ) b_{2}} \]

Problem 1946

ODE	\[ \boxed {x \left (6 y x +5\right )+\left (2 x^{3}+3 y\right ) y^{\prime }=0} \]
program solution	\[ 2 y x^{3}+\frac {5 x^{2}}{2}+\frac {3 y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 x^{3}}{3}-\frac {\sqrt {4 x^{6}-15 x^{2}-6 c_{1}}}{3} \\ y \left (x \right ) &= -\frac {2 x^{3}}{3}+\frac {\sqrt {4 x^{6}-15 x^{2}-6 c_{1}}}{3} \\ \end{align*}

Problem 1947

ODE	\[ \boxed {3 x^{2} y+x y^{2}+\left (x^{3}+x^{2} y+\sin \left (y\right )\right ) y^{\prime }=-{\mathrm e}^{x}} \]
program solution	\[ \frac {x^{2} y^{2}}{2}+y x^{3}+{\mathrm e}^{x}-\cos \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ \frac {x^{2} y \left (x \right )^{2}}{2}+x^{3} y \left (x \right )+{\mathrm e}^{x}-\cos \left (y \left (x \right )\right )+c_{1} = 0 \]

Problem 1948

ODE	\[ \boxed {2 y x -\left (x^{2}+y^{2}\right ) y^{\prime }=0} \]
program solution	\[ \frac {x^{2}}{y}-y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1-\sqrt {4 c_{1}^{2} x^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {4 c_{1}^{2} x^{2}+1}}{2 c_{1}} \\ \end{align*}

Problem 1949

ODE	\[ \boxed {\cos \left (x \right ) y-2 \sin \left (y\right )-\left (2 x \cos \left (y\right )-\sin \left (x \right )\right ) y^{\prime }=0} \]
program solution	\[ \sin \left (x \right ) y-2 \sin \left (y\right ) x = c_{1} \] Verified OK.
Maple solution	\[ \sin \left (x \right ) y \left (x \right )-2 x \sin \left (y \left (x \right )\right )+c_{1} = 0 \]

Problem 1950

ODE	\[ \boxed {\frac {2 y x -1}{y}+\frac {\left (x +3 y\right ) y^{\prime }}{y^{2}}=0} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {x^{2}}{3}-\frac {c_{1}}{3}}}{3}\right )-\frac {x^{2}}{3}+\frac {c_{1}}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{3 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {x^{2}}{3}} c_{1} x}{3}\right )} \]

Problem 1951

ODE	\[ \boxed {{\mathrm e}^{x} y+{\mathrm e}^{x} y^{\prime }=2 x} \]
program solution	\[ y = {\mathrm e}^{-x} \left (x^{2}+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{2}+c_{1} \right ) {\mathrm e}^{-x} \]

Problem 1952

ODE	\[ \boxed {3 \sin \left (x \right ) y-\cos \left (y\right )+\left (\sin \left (y\right ) x -3 \cos \left (x \right )\right ) y^{\prime }=0} \]
program solution	\[ -3 \cos \left (x \right ) y-x \cos \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ -3 \cos \left (x \right ) y \left (x \right )-x \cos \left (y \left (x \right )\right )+c_{1} = 0 \]

Problem 1953

ODE	\[ \boxed {x y^{2}+2 y+\left (2 y^{3}-x^{2} y+2 x \right ) y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1954

ODE	\[ \boxed {\frac {2}{y}-\frac {y}{x^{2}}+\left (\frac {1}{x}-\frac {2 x}{y^{2}}\right ) y^{\prime }=0} \]
program solution	\[ y = x \,{\mathrm e}^{c_{1}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {2}\, x \\ y \left (x \right ) &= -\sqrt {2}\, x \\ y \left (x \right ) &= -\frac {\left (c_{1} +\sqrt {c_{1}^{2}-8}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (-c_{1} +\sqrt {c_{1}^{2}-8}\right ) x}{2} \\ \end{align*}

Problem 1955

ODE	\[ \boxed {\frac {y x +1}{y}+\frac {\left (-x +2 y\right ) y^{\prime }}{y^{2}}=0} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}^{\frac {x^{2}}{4}-\frac {c_{1}}{2}}}{2}\right )-\frac {x^{2}}{4}+\frac {c_{1}}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2 \operatorname {LambertW}\left (-\frac {x c_{1} {\mathrm e}^{\frac {x^{2}}{4}}}{2}\right )} \]

Problem 1956

ODE	\[ \boxed {\frac {y \left (2+y x^{3}\right )}{x^{3}}-\frac {\left (1-2 y x^{3}\right ) y^{\prime }}{x^{2}}=0} \]
program solution	\[ \frac {y \left (y x^{3}-1\right )}{x^{2}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {c_{1}^{3}-\sqrt {c_{1}^{6}+4 c_{1} x^{5}}}{2 x^{3} c_{1}^{3}} \\ y \left (x \right ) &= \frac {c_{1}^{3}+\sqrt {c_{1}^{6}+4 c_{1} x^{5}}}{2 c_{1}^{3} x^{3}} \\ \end{align*}

Problem 1957

ODE	\[ \boxed {y^{2} \csc \left (x \right )^{2}+6 y x -\left (2 y \cot \left (x \right )-3 x^{2}\right ) y^{\prime }=2} \]
program solution	\[ -2 x -y^{2} \cot \left (x \right )+3 x^{2} y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 \tan \left (x \right ) x^{2}}{2}-\frac {\sqrt {\tan \left (x \right ) \left (9 \tan \left (x \right ) x^{4}+4 c_{1} -8 x \right )}}{2} \\ y \left (x \right ) &= \frac {3 \tan \left (x \right ) x^{2}}{2}+\frac {\sqrt {\tan \left (x \right ) \left (9 \tan \left (x \right ) x^{4}+4 c_{1} -8 x \right )}}{2} \\ \end{align*}

Problem 1958

ODE	\[ \boxed {\frac {2 y}{x^{3}}+\frac {2 x}{y^{2}}-\left (\frac {1}{x^{2}}+\frac {2 x^{2}}{y^{3}}\right ) y^{\prime }=0} \]
program solution	\[ -\frac {y}{x^{2}}+\frac {x^{2}}{y^{2}} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{\frac {4}{3}} \operatorname {RootOf}\left (x^{14} \textit {\_Z}^{18}+6 x^{\frac {38}{3}} \textit {\_Z}^{16}+15 x^{\frac {34}{3}} \textit {\_Z}^{14}+\left (20 x^{10}-27 c_{1} x^{8}\right ) \textit {\_Z}^{12}+\left (15 x^{\frac {26}{3}}-81 x^{\frac {20}{3}} c_{1} \right ) \textit {\_Z}^{10}+\left (6 x^{\frac {22}{3}}-108 x^{\frac {16}{3}} c_{1} \right ) \textit {\_Z}^{8}+\left (x^{6}-81 c_{1} x^{4}\right ) \textit {\_Z}^{6}-36 x^{\frac {8}{3}} c_{1} \textit {\_Z}^{4}-9 x^{\frac {4}{3}} c_{1} \textit {\_Z}^{2}-c_{1} \right )^{2}+1}{\operatorname {RootOf}\left (x^{14} \textit {\_Z}^{18}+6 x^{\frac {38}{3}} \textit {\_Z}^{16}+15 x^{\frac {34}{3}} \textit {\_Z}^{14}+\left (20 x^{10}-27 c_{1} x^{8}\right ) \textit {\_Z}^{12}+\left (15 x^{\frac {26}{3}}-81 x^{\frac {20}{3}} c_{1} \right ) \textit {\_Z}^{10}+\left (6 x^{\frac {22}{3}}-108 x^{\frac {16}{3}} c_{1} \right ) \textit {\_Z}^{8}+\left (x^{6}-81 c_{1} x^{4}\right ) \textit {\_Z}^{6}-36 x^{\frac {8}{3}} c_{1} \textit {\_Z}^{4}-9 x^{\frac {4}{3}} c_{1} \textit {\_Z}^{2}-c_{1} \right )^{2}} \]

Problem 1959

ODE	\[ \boxed {\cos \left (y\right )-\left (\sin \left (y\right ) x -y^{2}\right ) y^{\prime }=0} \]
program solution	\[ x \cos \left (y\right )+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\[ x +\frac {\sec \left (y \left (x \right )\right ) \left (y \left (x \right )^{3}-3 c_{1} \right )}{3} = 0 \]

Problem 1960

ODE	\[ \boxed {2 y \sin \left (y x \right )+\left (2 x \sin \left (y x \right )+y^{3}\right ) y^{\prime }=0} \]
program solution	\[ -2 \cos \left (y x \right )+\frac {y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-8 \cos \left (\textit {\_Z} \right ) x^{4}+4 c_{1} x^{4}+\textit {\_Z}^{4}\right )}{x} \]

Problem 1961

ODE	\[ \boxed {\frac {x \cos \left (\frac {x}{y}\right )}{y}+\sin \left (\frac {x}{y}\right )-\frac {x^{2} \cos \left (\frac {x}{y}\right ) y^{\prime }}{y^{2}}=-\cos \left (x \right )} \]
program solution	\[ x \sin \left (\frac {x}{y}\right )+\sin \left (x \right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{\arcsin \left (\frac {\sin \left (x \right )+c_{1}}{x}\right )} \]

Problem 1962

ODE	\[ \boxed {{\mathrm e}^{y x} y+2 y x +\left ({\mathrm e}^{y x} x +x^{2}\right ) y^{\prime }=0} \]
program solution	\[ y = \frac {-\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_{1}}{x}}}{x}\right )+\frac {c_{1}}{x}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-x \operatorname {LambertW}\left (\frac {{\mathrm e}^{-\frac {c_{1}}{x}}}{x}\right )-c_{1}}{x^{2}} \]

Problem 1963

ODE	\[ \boxed {\frac {x^{2}+3 y^{2}}{x \left (3 x^{2}+4 y^{2}\right )}+\frac {\left (y^{2}+2 x^{2}\right ) y^{\prime }}{y \left (3 x^{2}+4 y^{2}\right )}=0} \]
program solution	\[ -\frac {5 \ln \left (3 x^{2}+4 y^{2}\right )}{24}+\frac {2 \ln \left (y\right )}{3}+\frac {3 \ln \left (x \right )}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (x^{3} {\mathrm e}^{3 c_{1}} \textit {\_Z}^{24}-4 \textit {\_Z}^{15}-3 x^{3} {\mathrm e}^{3 c_{1}}\right )^{5} \sqrt {\frac {\operatorname {RootOf}\left (x^{3} {\mathrm e}^{3 c_{1}} \textit {\_Z}^{24}-4 \textit {\_Z}^{15}-3 x^{3} {\mathrm e}^{3 c_{1}}\right )^{5}}{x}}\, {\mathrm e}^{-\frac {3 c_{1}}{2}} \]

Problem 1964

ODE	\[ \boxed {\frac {x^{2}-y^{2}}{x \left (y^{2}+2 x^{2}\right )}+\frac {\left (2 y^{2}+x^{2}\right ) y^{\prime }}{y \left (y^{2}+2 x^{2}\right )}=0} \]
program solution	\[ \frac {3 \ln \left (y^{2}+2 x^{2}\right )}{4}-\ln \left (x \right )+\frac {\ln \left (y\right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \operatorname {RootOf}\left (\textit {\_Z}^{16} c_{1}^{2}+2 x^{4} \textit {\_Z}^{4}-x^{4}\right )^{6}}{x} \]

Problem 1965

ODE	\[ \boxed {\frac {2 x^{2}}{x^{2}+y^{2}}+\ln \left (x^{2}+y^{2}\right )+\frac {2 x y y^{\prime }}{x^{2}+y^{2}}=0} \]
program solution	\[ \ln \left (x^{2}+y^{2}\right ) x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {-x^{2}+{\mathrm e}^{-\frac {c_{1}}{x}}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+{\mathrm e}^{-\frac {c_{1}}{x}}} \\ \end{align*}

Problem 1966

ODE	\[ \boxed {-y+y^{\prime } x=-\ln \left (x \right )} \]
program solution	\[ y = c_{1} x +\ln \left (x \right )+1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +\ln \left (x \right )+1 \]

Problem 1967

ODE	\[ \boxed {y x +\left (x^{2}+y\right ) y^{\prime }=0} \]
program solution	\[ \frac {x^{2} y^{2}}{2}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-1+\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x^{2}}{2} \\ y \left (x \right ) &= \frac {\left (-1+\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x^{2}}{2} \\ y \left (x \right ) &= \frac {\left (-1+\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x^{2}}{2} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}-x^{4}-2 x^{2} \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}-\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x^{2}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}-x^{4}-2 x^{2} \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}-\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x^{2}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}-x^{4}-2 x^{2} \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}-\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x^{2}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 1968

ODE	\[ \boxed {\left (-2 y x +x \right ) y^{\prime }+2 y=0} \]
program solution	\[ y = -\frac {\operatorname {LambertW}\left (-\frac {2 \,{\mathrm e}^{-2 c_{1}}}{x^{2}}\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\operatorname {LambertW}\left (-\frac {2 c_{1}}{x^{2}}\right )}{2} \]

Problem 1969

ODE	\[ \boxed {x^{2} y+y^{2}+y^{\prime } x^{3}=0} \]
program solution	\[ y = -\frac {3}{x \left (c_{3} +\frac {1}{x^{3}}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 x^{2}}{3 c_{1} x^{3}-1} \]

Problem 1970

ODE	\[ \boxed {x y^{3}+x^{2} y^{2} y^{\prime }=1} \]
program solution	\[ \frac {x^{3} y^{3}}{3}-\frac {x^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (12 x^{2}+8 c_{1} \right )^{\frac {1}{3}}}{2 x} \\ y \left (x \right ) &= -\frac {\left (12 x^{2}+8 c_{1} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y \left (x \right ) &= \frac {\left (12 x^{2}+8 c_{1} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

Problem 1971

ODE	\[ \boxed {\left (x^{3} y^{3}-1\right ) y^{\prime }+x^{2} y^{4}=0} \]
program solution	\[ y = {\mathrm e}^{-\frac {\operatorname {LambertW}\left (-x^{3} {\mathrm e}^{-3 c_{1}}\right )}{3}-c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-c_{1}}}{\left (-\frac {x^{3} {\mathrm e}^{-3 c_{1}}}{\operatorname {LambertW}\left (-x^{3} {\mathrm e}^{-3 c_{1}}\right )}\right )^{\frac {1}{3}}} \]

Problem 1972

ODE	\[ \boxed {y \left (-x^{2}+y\right )+y^{\prime } x^{3}=0} \]
program solution	\[ y = -\frac {x}{c_{3} +\frac {1}{x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}}{c_{1} x -1} \]

Problem 1973

ODE	\[ \boxed {y+x y^{2}+\left (x -x^{2} y\right ) y^{\prime }=0} \]
program solution	\[ y = -\frac {1}{x \operatorname {LambertW}\left (-\frac {{\mathrm e}^{c_{1}}}{x^{2}}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{\operatorname {LambertW}\left (-\frac {c_{1}}{x^{2}}\right ) x} \]

Problem 1974

ODE	\[ \boxed {\left (x -x \sqrt {x^{2}-y^{2}}\right ) y^{\prime }-y=0} \]
program solution
Maple solution	\[ y \left (x \right )-\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )-c_{1} = 0 \]

Problem 1975

ODE	\[ \boxed {2 y x +\left (-x^{2}+y\right ) y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-x^{2} {\mathrm e}^{-c_{1}}\right )+c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x^{2}}{\operatorname {LambertW}\left (-c_{1} x^{2}\right )} \]

Problem 1976

ODE	\[ \boxed {y-x \left (x^{2} y-1\right ) y^{\prime }=0} \]
program solution	\[ -\frac {1}{2 x^{2} y^{2}}+\frac {1}{y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x +\sqrt {x^{2}-c_{1}}}{c_{1} x} \\ y \left (x \right ) &= \frac {x -\sqrt {x^{2}-c_{1}}}{c_{1} x} \\ \end{align*}

Problem 1977

ODE	\[ \boxed {{\mathrm e}^{x} y^{\prime }-2 x y^{2}-{\mathrm e}^{x} y=0} \]
program solution	\[ y = -\frac {{\mathrm e}^{x}}{x^{2}+c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x}}{-x^{2}+c_{1}} \]

Problem 1978

ODE	\[ \boxed {\left (x^{2}+y^{2}+x \right ) y^{\prime }-y=0} \]
program solution	\[ -\arctan \left (\frac {x}{y}\right )+y = c_{1} \] Verified OK.
Maple solution	\[ \frac {{\mathrm e}^{-2 i y \left (x \right )} \left (i x +y \left (x \right )\right )+2 c_{1} \left (x +i y \left (x \right )\right )}{2 i y \left (x \right )+2 x} = 0 \]

Problem 1979

ODE	\[ \boxed {\left (2 x +3 x^{2} y\right ) y^{\prime }+y+2 x y^{2}=0} \]
program solution	\[ y^{2} x \left (y x +1\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2 c_{1}^{2} 2^{\frac {1}{3}}-2 c_{1} \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {2}{3}}}{6 c_{1} x \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2 \left (i \sqrt {3}-1\right ) c_{1}^{2} 2^{\frac {1}{3}}-\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) 2^{\frac {2}{3}}+4 c_{1} \right )}{12 \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} c_{1} x} \\ y \left (x \right ) &= \frac {-2 \left (1+i \sqrt {3}\right ) c_{1}^{2} 2^{\frac {1}{3}}+\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}}-4 c_{1} \right )}{12 \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} c_{1} x} \\ \end{align*}

Problem 1980

ODE	\[ \boxed {2 y^{\prime } x^{2} y-2 x y^{2}=-{\mathrm e}^{x} x^{4}} \]
program solution	\[ \frac {x^{2} {\mathrm e}^{x}+y^{2}}{x^{2}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {-{\mathrm e}^{x}+c_{1}}\, x \\ y \left (x \right ) &= -\sqrt {-{\mathrm e}^{x}+c_{1}}\, x \\ \end{align*}

Problem 1981

ODE	\[ \boxed {y \left (1-x^{4} y^{2}\right )+y^{\prime } x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
program solution	\[ \frac {-x^{4} y^{2}-1}{2 x^{2} y^{2}} = -1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{\sqrt {-x^{2}+2}\, x} \]

Problem 1982

ODE	\[ \boxed {y \left (x^{2}-1\right )+x \left (x^{2}+1\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ y = \frac {4 x}{x^{2}+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 x}{x^{2}+1} \]

Problem 1983

ODE	\[ \boxed {x^{2} y^{2}-y+\left (2 y x^{3}+x \right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (2\right ) = -2] \end {align*}
program solution	\[ \frac {y \left (x^{2} y+1\right )}{x} = 7 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-1-\sqrt {28 x^{3}+1}}{2 x^{2}} \]

Problem 1984

ODE	\[ \boxed {\left (x^{2}+y^{2}-2 y\right ) y^{\prime }=2 x} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ {\mathrm e}^{-y} \left (-y^{2}-x^{2}\right ) = -1 \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 1985

ODE	\[ \boxed {y-x^{2} \sqrt {x^{2}-y^{2}}-y^{\prime } x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1986

ODE	\[ \boxed {y \left (x +y^{2}\right )+x \left (-y^{2}+x \right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 2] \end {align*}
program solution	\[ -\frac {2 \ln \left (y^{2}+3 x \right )}{3}+\frac {\ln \left (y\right )}{3} = -\ln \left (x \right )+\frac {2 \ln \left (2\right )}{3}-\frac {2 \ln \left (5\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-3 \ln \left (x \right )+4 \ln \left (2\right )-4 \ln \left (5\right )+4 \ln \left (\frac {\textit {\_Z}^{2}+3 x}{x}\right )-2 \ln \left (\frac {\textit {\_Z}}{\sqrt {x}}\right )\right ) \]

Problem 1987

ODE	\[ \boxed {y^{\prime } x +2 y=x^{2}} \]
program solution	\[ y = \frac {x^{4}+4 c_{1}}{4 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{4}+4 c_{1}}{4 x^{2}} \]

Problem 1988

ODE	\[ \boxed {-y x +y^{\prime }={\mathrm e}^{\frac {x^{2}}{2}} \cos \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{\frac {x^{2}}{2}} \left (\sin \left (x \right )+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\sin \left (x \right )+c_{1} \right ) {\mathrm e}^{\frac {x^{2}}{2}} \]

Problem 1989

ODE	\[ \boxed {2 y x +y^{\prime }=2 x \,{\mathrm e}^{-x^{2}}} \]
program solution	\[ y = {\mathrm e}^{-x^{2}} \left (x^{2}+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{2}+c_{1} \right ) {\mathrm e}^{-x^{2}} \]

Problem 1990

ODE	\[ \boxed {y^{\prime }-y=3 x^{2} {\mathrm e}^{x}} \]
program solution	\[ y = {\mathrm e}^{x} \left (x^{3}+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{3}+c_{1} \right ) {\mathrm e}^{x} \]

Problem 1991

ODE	\[ \boxed {x^{\prime }+x={\mathrm e}^{-y}} \]
program solution	\[ x = {\mathrm e}^{-y} \left (y +c_{1} \right ) \] Verified OK.
Maple solution	\[ x \left (y \right ) = \left (y +c_{1} \right ) {\mathrm e}^{-y} \]

Problem 1992

ODE	\[ \boxed {y x^{\prime }+\left (y +1\right ) x={\mathrm e}^{y}} \]
program solution	\[ x = \frac {\left ({\mathrm e}^{2 y}+2 c_{1} \right ) {\mathrm e}^{-y}}{2 y} \] Verified OK.
Maple solution	\[ x \left (y \right ) = \frac {{\mathrm e}^{y}+2 \,{\mathrm e}^{-y} c_{1}}{2 y} \]

Problem 1993

ODE	\[ \boxed {y+\left (-3 y+2 x \right ) y^{\prime }=0} \]
program solution	\[ x y^{2}-y^{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {c_{1} \left (-4 x^{3}+27 c_{1} \right )}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{3 \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {c_{1} \left (-4 x^{3}+27 c_{1} \right )}\right )^{\frac {1}{3}}}+\frac {x}{3} \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\left (i \sqrt {3}\, x -x +\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x}{3 \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {\left (i \sqrt {3}\, x +x -\left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x}{3 \left (-108 c_{1} +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+27 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 1994

ODE	\[ \boxed {y^{\prime } x -2 y=2 x^{4}} \]
program solution	\[ y = x^{2} \left (x^{2}+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{2}+c_{1} \right ) x^{2} \]

Problem 1995

ODE	\[ \boxed {-\left ({\mathrm e}^{y}+x \right ) y^{\prime }=-1} \]
program solution	\[ y = \operatorname {LambertW}\left (x \,{\mathrm e}^{c_{1}}\right )-c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {LambertW}\left (x \,{\mathrm e}^{c_{1}}\right )-c_{1} \]

Problem 1996

ODE	\[ \boxed {y^{2} x^{\prime }+\left (y^{2}+2 y \right ) x=1} \]
program solution	\[ x = \frac {\left ({\mathrm e}^{y}+c_{1} \right ) {\mathrm e}^{-y}}{y^{2}} \] Verified OK.
Maple solution	\[ x \left (y \right ) = \frac {{\mathrm e}^{-y} c_{1} +1}{y^{2}} \]

Problem 1997

ODE	\[ \boxed {y^{\prime } x -5 y=x +1} \]
program solution	\[ y = -\frac {1}{5}-\frac {1}{4} x +c_{1} x^{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{5}-\frac {1}{4} x +c_{1} x^{5} \]

Problem 1998

ODE	\[ \boxed {y^{\prime } x^{2}+y-2 y x=2 x^{2}} \]
program solution	\[ y = x^{2} \left (2 \,{\mathrm e}^{-\frac {1}{x}}+c_{1} \right ) {\mathrm e}^{\frac {1}{x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (2+c_{1} {\mathrm e}^{\frac {1}{x}}\right ) \]

Problem 1999

ODE	\[ \boxed {\left (x +1\right ) y^{\prime }+2 y=\frac {{\mathrm e}^{x}}{x +1}} \]
program solution	\[ y = \frac {{\mathrm e}^{x}+c_{1}}{x^{2}+2 x +1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x}+c_{1}}{\left (x +1\right )^{2}} \]

Problem 2000

ODE	\[ \boxed {\cos \left (y\right )^{2}+\left (x -\tan \left (y\right )\right ) y^{\prime }=0} \]
program solution	\[ {\mathrm e}^{\tan \left (y\right )} x -\tan \left (y\right ) {\mathrm e}^{\tan \left (y\right )}+{\mathrm e}^{\tan \left (y\right )} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arctan \left (\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-x -1}\right )+x +1\right ) \]