2.17.35 Problems 3401 to 3500

Problem 3401

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

Problem 3402

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

Problem 3403

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

Problem 3404

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

Problem 3405

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

Problem 3406

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

Problem 3407

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

Problem 3408

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

Problem 3409

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

Problem 3410

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

Problem 3411

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

Problem 3412

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

Problem 3413

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

Problem 3414

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

Problem 3415

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

Problem 3416

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

Problem 3417

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

Problem 3418

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

Problem 3419

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

Problem 3420

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

Problem 3421

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

Problem 3422

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

Problem 3423

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

Problem 3424

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

Problem 3425

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

Problem 3426

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

Problem 3427

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

Problem 3428

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

Problem 3429

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

Problem 3430

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

Problem 3431

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

Problem 3432

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

Problem 3433

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

Problem 3434

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

Problem 3435

ODE	\[ \boxed {x y^{\prime }+\left (\operatorname {a2} +\operatorname {a3} x y\right ) y=-\operatorname {a1} x -\operatorname {a0}} \]
program solution	\[ y = \frac {\frac {\left (-\left (\frac {1}{2} \operatorname {a2}^{2}-\operatorname {a2} \right ) \operatorname {a1}^{\frac {3}{2}}+i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} -\frac {\operatorname {a0}^{2} \operatorname {a3} \sqrt {\operatorname {a1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}+\frac {\left (i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} +\operatorname {a1}^{\frac {3}{2}} \operatorname {a2} \right ) c_{3} \operatorname {KummerM}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}-\left (\frac {\operatorname {a1}^{\frac {3}{2}} \operatorname {a2}}{2}+i \sqrt {\operatorname {a3}}\, \operatorname {a1} \left (\operatorname {a1} x +\frac {\operatorname {a0}}{2}\right )\right ) \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )}{\operatorname {a1}^{\frac {3}{2}} x \operatorname {a3} \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {4 \operatorname {a1} \left (\operatorname {a1}^{3} \operatorname {a3} \left (\operatorname {a3} \operatorname {a0} -\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-\frac {c_{1} \left (\operatorname {a0}^{2} \operatorname {a3} +\operatorname {a1} \,\operatorname {a2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )}{4}+\operatorname {a1}^{3} \operatorname {a3} \left (\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}+\operatorname {a3} \operatorname {a0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) c_{1} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} -\operatorname {a1} \operatorname {a2} \right )}{2}\right )}{4 \operatorname {a1}^{3} \operatorname {a3}^{2} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-c_{1} \sqrt {-\operatorname {a1} \operatorname {a3}}\, \left (\operatorname {a0}^{2} \operatorname {a3} +\operatorname {a1} \,\operatorname {a2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-4 \left (\operatorname {a1}^{2} \operatorname {a3}^{2} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} -\operatorname {a1} \operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-\frac {\operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) c_{1} \left (\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}+\operatorname {a3} \operatorname {a0} \right )}{2}\right ) \operatorname {a1}} \]

Problem 3436

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

Problem 3437

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

Problem 3438

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

Problem 3439

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

Problem 3440

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

Problem 3441

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

Problem 3442

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

Problem 3443

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

Problem 3444

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

Problem 3445

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

Problem 3446

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

Problem 3447

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

Problem 3448

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

Problem 3449

ODE	\[ \boxed {x y^{\prime }+2 y-\sqrt {1+y^{2}}=0} \]
program solution	\[ -\ln \left (x \right )-\frac {\sqrt {3}\, \sqrt {9 \left (y-\frac {\sqrt {3}}{3}\right )^{2}+6 \sqrt {3}\, \left (y-\frac {\sqrt {3}}{3}\right )+12}}{18}-\frac {\operatorname {arcsinh}\left (y\right )}{3}+\frac {\operatorname {arctanh}\left (\frac {3 \left (\frac {8}{3}+\frac {2 \sqrt {3}\, \left (y-\frac {\sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{4 \sqrt {9 \left (y-\frac {\sqrt {3}}{3}\right )^{2}+6 \sqrt {3}\, \left (y-\frac {\sqrt {3}}{3}\right )+12}}\right )}{3}+\frac {\sqrt {3}\, \sqrt {9 \left (y+\frac {\sqrt {3}}{3}\right )^{2}-6 \sqrt {3}\, \left (y+\frac {\sqrt {3}}{3}\right )+12}}{18}-\frac {\operatorname {arctanh}\left (\frac {3 \left (\frac {8}{3}-\frac {2 \sqrt {3}\, \left (y+\frac {\sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{4 \sqrt {9 \left (y+\frac {\sqrt {3}}{3}\right )^{2}-6 \sqrt {3}\, \left (y+\frac {\sqrt {3}}{3}\right )+12}}\right )}{3}-\frac {\ln \left (3 y^{2}-1\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )-\left (\int _{}^{y \left (x \right )}\frac {1}{-2 \textit {\_a} +\sqrt {\textit {\_a}^{2}+1}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 3450

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

Problem 3451

ODE	\[ \boxed {x y^{\prime }-y-\sqrt {x^{2}-y^{2}}=0} \]
program solution	\[ y = -\tan \left (c_{1} -\ln \left (x \right )\right ) \sqrt {\frac {x^{2}}{\tan \left (c_{1} -\ln \left (x \right )\right )^{2}+1}} \] 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 3452

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

Problem 3453

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

Problem 3454

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

Problem 3455

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

Problem 3456

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

Problem 3457

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

Problem 3458

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

Problem 3459

ODE	\[ \boxed {x y^{\prime }-y+\cot \left (y\right )^{2}=0} \]
program solution	\[ -\ln \left (x \right )+\int _{0}^{y}\frac {1}{\textit {\_a} -\cot \left (\textit {\_a} \right )^{2}}d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )+c_{1} +\int _{}^{y \left (x \right )}\frac {1}{\cot \left (\textit {\_a} \right )^{2}-\textit {\_a}}d \textit {\_a} = 0 \]

Problem 3460

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

Problem 3461

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

Problem 3462

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

Problem 3463

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

Problem 3464

ODE	\[ \boxed {x y^{\prime }-y-x \sin \left (\frac {y}{x}\right )=0} \]
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 3465

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

Problem 3466

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

Problem 3467

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

Problem 3468

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

Problem 3469

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

Problem 3470

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

Problem 3471

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

Problem 3472

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

Problem 3473

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

Problem 3474

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

Problem 3475

ODE	\[ \boxed {x y^{\prime }+y n -f \left (x \right ) g \left (x^{n} y\right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (-\left (\int x^{n -1} f \left (x \right )d x \right )+\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x^{-n} \]

Problem 3476

ODE	\[ \boxed {x y^{\prime }-y f \left (x^{m} y^{n}\right )=0} \]
program solution	\[ -\frac {m \ln \left (x \right )}{n} = \int _{}^{y x^{\frac {m}{n}}}-\frac {m}{\textit {\_a} \left (n f \left (\textit {\_a}^{n}\right )+m \right )}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{\left (f \left (x^{m} \textit {\_a}^{n}\right ) n +m \right ) \textit {\_a}}d \textit {\_a} -\frac {\ln \left (x \right )}{n}-c_{1} = 0 \]

Problem 3477

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

Problem 3478

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

Problem 3479

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

Problem 3480

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

Problem 3481

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

Problem 3482

ODE	\[ \boxed {\left (x +1\right ) y^{\prime }-\left (1-x y^{3}\right ) y=0} \]
program solution	\[ y = \frac {2^{\frac {2}{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} \left (x +1\right )}{3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1}} \] Verified OK. \[ y = \frac {{\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} 2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (x +1\right )}{6 x^{4}+16 x^{3}+12 x^{2}+8 c_{1}} \] Verified OK. \[ y = -\frac {{\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} 2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (x +1\right )}{6 x^{4}+16 x^{3}+12 x^{2}+8 c_{1}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} \left (x +1\right )}{3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1}} \\ y \left (x \right ) &= -\frac {{\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} 2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (x +1\right )}{6 x^{4}+16 x^{3}+12 x^{2}+8 c_{1}} \\ y \left (x \right ) &= \frac {{\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} 2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (x +1\right )}{6 x^{4}+16 x^{3}+12 x^{2}+8 c_{1}} \\ \end{align*}

Problem 3483

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

Problem 3484

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

Problem 3485

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

Problem 3486

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

Problem 3487

ODE	\[ \boxed {\left (a +x \right ) y^{\prime }-3 y=2 \left (a +x \right )^{5}} \]
program solution	\[ y = \left (a +x \right )^{3} \left (2 x a +x^{2}+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 a x +x^{2}+c_{1} \right ) \left (x +a \right )^{3} \]

Problem 3488

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

Problem 3489

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

Problem 3490

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

Problem 3491

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

Problem 3492

ODE	\[ \boxed {2 x y^{\prime }+y=2 x^{3}} \]
program solution	\[ y = \frac {4 x^{\frac {7}{2}}+7 c_{1}}{14 \sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 x^{3}}{7}+\frac {c_{1}}{\sqrt {x}} \]

Problem 3493

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

Problem 3494

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

Problem 3495

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

Problem 3496

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

Problem 3497

ODE	\[ \boxed {2 x y^{\prime }+4 y+\sqrt {a^{2}-4 b -4 c y}=-a} \]
program solution	\[ -\ln \left (x \right )+\frac {\ln \left (-4 c y+c \sqrt {a^{2}-4 b -4 c y}-a c \right )}{4}+\frac {c \,\operatorname {arctanh}\left (\frac {c +2 \sqrt {a^{2}-4 b -4 c y}}{\sqrt {4 a^{2}+4 a c +c^{2}-16 b}}\right )}{2 \sqrt {4 a^{2}+4 a c +c^{2}-16 b}}-\frac {\ln \left (-4 c y-c \sqrt {a^{2}-4 b -4 c y}-a c \right )}{4}+\frac {c \,\operatorname {arctanh}\left (\frac {2 \sqrt {a^{2}-4 b -4 c y}-c}{\sqrt {4 a^{2}+4 a c +c^{2}-16 b}}\right )}{2 \sqrt {4 a^{2}+4 a c +c^{2}-16 b}}-\frac {\ln \left (2 a y+c y+4 y^{2}+b \right )}{4}-\frac {c \,\operatorname {arctanh}\left (\frac {2 a +c +8 y}{\sqrt {4 a^{2}+4 a c +c^{2}-16 b}}\right )}{2 \sqrt {4 a^{2}+4 a c +c^{2}-16 b}} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )+2 \left (\int _{}^{y \left (x \right )}\frac {1}{4 \textit {\_a} +a +\sqrt {-4 \textit {\_a} c +a^{2}-4 b}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 3498

ODE	\[ \boxed {\left (1-2 x \right ) y^{\prime }+6 y=32 x +16} \]
program solution	\[ y = -\frac {1}{2} c_{1} x^{3}+\frac {3}{4} c_{1} x^{2}-\frac {3}{8} c_{1} x +\frac {1}{16} c_{1} +8 x +\frac {4}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4}{3}+8 x +\left (2 x -1\right )^{3} c_{1} \]

Problem 3499

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

Problem 3500

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