2.17.36 Problems 3501 to 3600

Problem 3501

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

Problem 3502

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

Problem 3503

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

Problem 3504

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

Problem 3505

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

Problem 3506

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

Problem 3507

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

Problem 3508

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

Problem 3509

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

Problem 3510

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

Problem 3511

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

Problem 3512

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

Problem 3513

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

Problem 3514

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

Problem 3515

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

Problem 3516

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

Problem 3517

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

Problem 3518

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

Problem 3519

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

Problem 3520

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

Problem 3521

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

Problem 3522

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

Problem 3523

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

Problem 3524

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

Problem 3525

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

Problem 3526

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

Problem 3527

ODE	\[ \boxed {y^{\prime } x^{2}-2 y \left (x -y^{2}\right )=0} \]
program solution	\[ y = \frac {3 x^{2}}{\sqrt {12 x^{3}+9 c_{1}}} \] Verified OK. \[ y = -\frac {3 x^{2}}{\sqrt {12 x^{3}+9 c_{1}}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {3 x^{2}}{\sqrt {12 x^{3}+9 c_{1}}} \\ y \left (x \right ) &= \frac {3 x^{2}}{\sqrt {12 x^{3}+9 c_{1}}} \\ \end{align*}

Problem 3528

ODE	\[ \boxed {y^{\prime } x^{2}-a \,x^{2} y^{2}+a y^{3}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {1}{-a x -2^{\frac {2}{3}} \left (-a \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right )+\operatorname {AiryBi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right ) c_{1} +\operatorname {AiryAi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right )\right )} \]

Problem 3529

ODE	\[ \boxed {y^{\prime } x^{2}+a y^{2}+y^{3} b \,x^{2}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = -\frac {2^{\frac {1}{3}} a b x}{2^{\frac {1}{3}} a^{2} b -2 \left (a^{2} b^{2}\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (-\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (-\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right )+\operatorname {AiryBi}\left (1, -\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right )\right ) x} \]

Problem 3530

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

Problem 3531

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

Problem 3532

ODE	\[ \boxed {y^{\prime } x^{2}-\sec \left (y\right )-3 x \tan \left (y\right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \arcsin \left (\frac {c_{1} x^{4}-1}{4 x}\right ) \]

Problem 3533

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

Problem 3534

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

Problem 3535

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }+y x=5} \]
program solution	\[ y = -c_{1} \sqrt {x^{2}-1}+5 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x -1}\, \sqrt {x +1}\, c_{1} +5 x \]

Problem 3536

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

Problem 3537

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

Problem 3538

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

Problem 3539

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

Problem 3540

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

Problem 3541

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

Problem 3542

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

Problem 3543

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

Problem 3544

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

Problem 3545

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

Problem 3546

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

Problem 3547

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

Problem 3548

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

Problem 3549

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

Problem 3550

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

Problem 3551

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

Problem 3552

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

Problem 3553

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

Problem 3554

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-n \left (y^{2}-2 y x +1\right )=0} \]
program solution	\[ y = \frac {\operatorname {LegendreQ}\left (n , x\right )+\operatorname {LegendreP}\left (n , x\right ) c_{3}}{\operatorname {LegendreQ}\left (n -1, x\right )+\operatorname {LegendreP}\left (n -1, x\right ) c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n} \left (16 \left (x +1\right )^{2} \left (\left (x -\frac {1}{2}\right ) n +\frac {1}{2}-\frac {x}{2}\right ) \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [2-2 n \right ], -\frac {2}{x -1}\right ) c_{1} \left (\frac {x +1}{x -1}\right )^{-n}+\left (x -1\right ) \left (\left (x +1\right )^{2} n \left (\frac {x +1}{x -1}\right )^{n} \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right )-16 \left (\frac {\operatorname {HeunCPrime}\left (0, 2 n -1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) \left (x +1\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 n}}{8}+\operatorname {HeunCPrime}\left (0, -2 n +1, 0, 0, n^{2}-n +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} \right ) \left (x -1\right )\right )\right )}{\left (x +1\right )^{2} \left (8 c_{1} \operatorname {hypergeom}\left (\left [-n +1, -n +1\right ], \left [2-2 n \right ], -\frac {2}{x -1}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 n}+\left (\frac {x +1}{x -1}\right )^{2 n} \operatorname {hypergeom}\left (\left [n , n\right ], \left [2 n \right ], -\frac {2}{x -1}\right ) \left (x -1\right )\right ) n} \]

Problem 3555

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

Problem 3556

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

Problem 3557

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime }-y^{2}+2 x y \left (1+y^{2}\right )=1} \]
program solution
Maple solution	\[ c_{1} +\frac {x}{{\left (\frac {\left (x^{2}+1\right ) \left (y \left (x \right )^{2}+1\right )}{\left (-1+x y \left (x \right )\right )^{2}}\right )}^{\frac {1}{4}}}+\frac {\left (x +y \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x +y \left (x \right )\right )^{2}}{\left (-1+x y \left (x \right )\right )^{2}}\right )}{2 x y \left (x \right )-2} = 0 \]

Problem 3558

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\arctan \left (\frac {6 \sqrt {x^{2}+1}\, \left (x^{2} \sqrt {x^{2}+1}+\sqrt {x^{2}+1}+3 c_{1} \right )}{10+6 c_{1} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}, \frac {8+6 \left (-x^{2}-1\right ) c_{1} \sqrt {x^{2}+1}-x^{6}-3 x^{4}+6 x^{2}-9 c_{1}^{2}}{10+6 c_{1} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}\right )}{2} \]

Problem 3559

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

Problem 3560

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

Problem 3561

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

Problem 3562

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

Problem 3563

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

Problem 3564

ODE	\[ \boxed {\left (a^{2}+x^{2}\right ) y^{\prime }-3 y x +2 y^{2}=a^{2}} \]
program solution	\[ y = \frac {\left (2 c_{3} x \sqrt {a^{2}+x^{2}}+a^{2}+2 x^{2}\right ) \sqrt {a^{2}+x^{2}}}{2 c_{3} a^{2}+2 c_{3} x^{2}+2 \sqrt {a^{2}+x^{2}}\, x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {2 \left (\sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, c_{1} a^{2} \left (i a -x \right ) \operatorname {HeunCPrime}\left (0, -\frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\sqrt {\frac {i x +a}{a}}\, a^{2} \left (i a -x \right ) \operatorname {HeunCPrime}\left (0, \frac {1}{2}, 2, 0, \frac {5}{4}, \frac {-i a +x}{i a +x}\right )+\frac {\sqrt {2}\, x \left (i a x -\frac {1}{2} a^{2}+\frac {1}{2} x^{2}\right ) c_{1} \sqrt {\frac {i x -a}{a}}}{2}-\frac {\sqrt {\frac {i x +a}{a}}\, \left (i a^{3}-3 i a \,x^{2}+3 x \,a^{2}-x^{3}\right )}{4}\right ) a}{\left (i \sqrt {2}\, \sqrt {\frac {i x -a}{a}}\, c_{1} x +\frac {\sqrt {\frac {i x +a}{a}}\, \left (i x -a \right )}{2}\right ) \left (i a +x \right )^{2}} \]

Problem 3565

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

Problem 3566

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

Problem 3567

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

Problem 3568

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

Problem 3569

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

Problem 3570

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime }+\left (2 x +1\right ) y=a} \]
program solution	\[ y = -\frac {3 c_{1} x^{3}-9 c_{1} x^{2}+9 c_{1} x -a -3 c_{1}}{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \left (x -1\right )^{3} c_{1} +a}{3 x} \]

Problem 3571

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime }-2 \left (-x +2\right ) y=a} \]
program solution	\[ y = -\frac {12 c_{1} x^{4}-4 x a +3 a}{12 \left (x^{2}-2 x +1\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {12 c_{1} x^{4}+4 a x -3 a}{12 \left (x -1\right )^{2}} \]

Problem 3572

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

Problem 3573

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

Problem 3574

ODE	\[ \boxed {\left (x -2\right ) \left (x -3\right ) y^{\prime }-8 y+3 y x=-x^{2}} \]
program solution	\[ y = \frac {-3 x^{4}+8 x^{3}+12 c_{1}}{12 x^{3}-84 x^{2}+192 x -144} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\frac {1}{4} x^{4}+\frac {2}{3} x^{3}+c_{1}}{\left (x -3\right ) \left (-2+x \right )^{2}} \]

Problem 3575

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

Problem 3576

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

Problem 3577

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

Problem 3578

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

Problem 3579

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

Problem 3580

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

Problem 3581

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

Problem 3582

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

Problem 3583

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

Problem 3584

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

Problem 3585

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

Problem 3586

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

Problem 3587

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

Problem 3588

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

Problem 3589

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

Problem 3590

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

Problem 3591

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

Problem 3592

ODE	\[ \boxed {2 \left (x^{2}+x +1\right ) y^{\prime }+\left (2 x +1\right ) y=8 x^{2}+1} \]
program solution	\[ y = \frac {4 x \sqrt {x^{2}+x +1}-6 \sqrt {x^{2}+x +1}+c_{1}}{2 \sqrt {x^{2}+x +1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 x -3+\frac {c_{1}}{\sqrt {x^{2}+x +1}} \]

Problem 3593

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

Problem 3594

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

Problem 3595

ODE	\[ \boxed {\left (b \,x^{2}+a \right ) y^{\prime }-B y^{2}=A} \]
program solution	\[ y = \frac {\left (-c_{3} \cos \left (\frac {\sqrt {A}\, \sqrt {B}\, \arctan \left (\frac {x b}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )+\sin \left (\frac {\sqrt {A}\, \sqrt {B}\, \arctan \left (\frac {x b}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\right ) \sqrt {A}}{\left (c_{3} \sin \left (\frac {\sqrt {A}\, \sqrt {B}\, \arctan \left (\frac {x b}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )+\cos \left (\frac {\sqrt {A}\, \sqrt {B}\, \arctan \left (\frac {x b}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )\right ) \sqrt {B}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\tan \left (\frac {\sqrt {A B}\, \left (c_{1} \sqrt {a b}+\arctan \left (\frac {x b}{\sqrt {a b}}\right )\right )}{\sqrt {a b}}\right ) \sqrt {A B}}{B} \]

Problem 3596

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

Problem 3597

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

Problem 3598

ODE	\[ \boxed {\left (b x +a \right )^{2} y^{\prime }+c y^{2}+\left (b x +a \right ) y^{3}=0} \]
program solution
Maple solution	\[ \frac {\left (\sqrt {b}\, a +b^{\frac {3}{2}} x \right ) {\mathrm e}^{-\frac {\left (\left (b x +a +c \right ) y \left (x \right )+b \left (b x +a \right )\right ) \left (\left (-b x -a +c \right ) y \left (x \right )+b \left (b x +a \right )\right )}{2 y \left (x \right )^{2} \left (b x +a \right )^{2} b}}+\frac {c \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {1}{2 b}} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (c y \left (x \right )+b \left (b x +a \right )\right )}{2 \sqrt {b}\, y \left (x \right ) \left (b x +a \right )}\right )}{2}+b^{\frac {3}{2}} c_{1}}{b^{\frac {3}{2}}} = 0 \]

Problem 3599

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

Problem 3600

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