2.17.34 Problems 3301 to 3400

Problem 3301

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

Problem 3302

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

Problem 3303

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

Problem 3304

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

Problem 3305

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

Problem 3306

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

Problem 3307

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

Problem 3308

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

Problem 3309

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

Problem 3310

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

Problem 3311

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

Problem 3312

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

Problem 3313

ODE	\[ \boxed {y^{\prime }-\left (\sin \left (2 x \right )+y\right ) y=\cos \left (2 x \right )} \]
program solution	\[ y = \frac {\left (2 \operatorname {HeunCPrime}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right )^{2}+2 c_{3} \cos \left (x \right ) \operatorname {HeunCPrime}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right ) \sin \left (x \right )}{c_{3} \operatorname {HeunC}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\cos \left (x \right ) \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (x \right ) \left (\operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} +2 \cos \left (x \right ) \left (\cos \left (x \right ) \operatorname {HeunCPrime}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} +\operatorname {HeunCPrime}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )\right )}{c_{1} \cos \left (x \right ) \operatorname {HeunC}\left (1, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\operatorname {HeunC}\left (1, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )} \]

Problem 3314

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

Problem 3315

ODE	\[ \boxed {y^{\prime }-\left (3+x -4 y\right )^{2}=0} \]
program solution	\[ y = \frac {c_{3} \left (2 x +5\right ) {\mathrm e}^{-2 x \left (x +5\right )}+{\mathrm e}^{-2 x \left (x +7\right )} \left (2 x +7\right )}{8 c_{3} {\mathrm e}^{-2 x \left (x +5\right )}+8 \,{\mathrm e}^{-2 x \left (x +7\right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (5+2 x \right ) {\mathrm e}^{4 x}-2 x -7}{8 \,{\mathrm e}^{4 x} c_{1} -8} \]

Problem 3316

ODE	\[ \boxed {y^{\prime }-\left (1+4 x +9 y\right )^{2}=0} \]
program solution	\[ y = \frac {\left (-2+\left (-12 x -3\right ) c_{3} \right ) \cos \left (6 x \right )-12 \sin \left (6 x \right ) \left (-\frac {c_{3}}{6}+\frac {1}{4}+x \right )}{27 c_{3} \cos \left (6 x \right )+27 \sin \left (6 x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {4 x}{9}-\frac {1}{9}-\frac {2 \tan \left (-6 x +6 c_{1} \right )}{27} \]

Problem 3317

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

Problem 3318

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

Problem 3319

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

Problem 3320

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

Problem 3321

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

Problem 3322

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

Problem 3323

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

Problem 3324

ODE	\[ \boxed {y^{\prime }-a y-b y^{2}=f \left (x \right )} \]
program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{b \operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 3325

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

Problem 3326

ODE	\[ \boxed {y^{\prime }-g \left (x \right ) y-a y^{2}=f \left (x \right )} \]
program solution	\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{f \left (x \right ) a \textit {\_Y} \left (x \right )-g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{a \operatorname {DESol}\left (\left \{f \left (x \right ) a \textit {\_Y} \left (x \right )-g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 3327

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

Problem 3328

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

Problem 3329

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

Problem 3330

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

Problem 3331

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

Problem 3332

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

Problem 3333

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

Problem 3334

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

Problem 3335

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

Problem 3336

ODE	\[ \boxed {y^{\prime }-\left (3-\cot \left (x \right )\right ) y-y^{2} \sin \left (x \right )=-4 \csc \left (x \right )} \]
program solution	\[ y = -\frac {3 \csc \left (x \right ) \left (\frac {8 \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{-\frac {5 i}{2}}}{3}-\frac {2 c_{3} \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{\frac {5 i}{2}}}{3}\right )}{2 c_{3} \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{\frac {5 i}{2}}+2 \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{-\frac {5 i}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 \csc \left (x \right ) \left (c_{1} \left (\operatorname {csgn}\left (\sin \left (x \right )\right )+\frac {5}{3}\right ) \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{-\frac {5 i}{2}}+\left (\cos \left (x \right )+i \sin \left (x \right )\right )^{\frac {5 i}{2}} \left (\operatorname {csgn}\left (\sin \left (x \right )\right )-\frac {5}{3}\right )\right )}{2 c_{1} \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{-\frac {5 i}{2}}+2 \left (\cos \left (x \right )+i \sin \left (x \right )\right )^{\frac {5 i}{2}}} \]

Problem 3337

ODE	\[ \boxed {y^{\prime }-y \sec \left (x \right )=\left (\sin \left (x \right )-1\right )^{2}} \]
program solution	\[ y = \frac {6 \tan \left (\frac {x}{2}\right )+4 \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right ) \tan \left (\frac {x}{2}\right )^{5}-8 \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \tan \left (\frac {x}{2}\right )^{5}+4 \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right ) \tan \left (\frac {x}{2}\right )^{4}-8 \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \tan \left (\frac {x}{2}\right )^{4}+8 \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right ) \tan \left (\frac {x}{2}\right )^{3}-16 \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \tan \left (\frac {x}{2}\right )^{3}+8 \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}-16 \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \tan \left (\frac {x}{2}\right )^{2}+4 \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right ) \tan \left (\frac {x}{2}\right )-8 \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \tan \left (\frac {x}{2}\right )-c_{1} +4 \ln \left (\sec \left (\frac {x}{2}\right )^{2}\right )-c_{1} \tan \left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )^{3}-8 \ln \left (\tan \left (\frac {x}{2}\right )+1\right )+6 \tan \left (\frac {x}{2}\right )^{4}+4 \tan \left (\frac {x}{2}\right )^{2}-\tan \left (\frac {x}{2}\right )^{5} c_{1} -\tan \left (\frac {x}{2}\right )^{4} c_{1} -2 \tan \left (\frac {x}{2}\right )^{3} c_{1} -2 \tan \left (\frac {x}{2}\right )^{2} c_{1}}{-1-\tan \left (\frac {x}{2}\right )^{4}-2 \tan \left (\frac {x}{2}\right )^{2}+2 \tan \left (\frac {x}{2}\right )^{3}+\tan \left (\frac {x}{2}\right )+\tan \left (\frac {x}{2}\right )^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-3 \sin \left (x \right )+4 \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+4 \ln \left (\cos \left (x \right )\right )-\frac {\cos \left (2 x \right )}{4}+c_{1} \right ) \left (\sec \left (x \right )+\tan \left (x \right )\right ) \]

Problem 3338

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

Problem 3339

ODE	\[ \boxed {y^{\prime }-g \left (x \right ) y-h \left (x \right ) y^{2}=f \left (x \right )} \]
program solution	\[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{h \left (x \right ) \operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 3340

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

Problem 3341

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

Problem 3342

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

Problem 3343

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

Problem 3344

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

Problem 3345

ODE	\[ \boxed {y^{\prime }-\operatorname {a1} y-\operatorname {a2} y^{2}-\operatorname {a3} y^{3}=\operatorname {a0}} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3} \operatorname {a3} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple solution	\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a}^{3} \operatorname {a3} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 3346

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

Problem 3347

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

Problem 3348

ODE	\[ \boxed {y^{\prime }-\left (a +b x y\right ) y^{2}=0} \]
program solution	\[ \frac {2 \ln \left (3 \left (y-\frac {a}{3 b x}\right ) b x -a \right ) \sqrt {a^{2}-4 b}-\ln \left (9 b^{2} x^{2} \left (y-\frac {a}{3 b x}\right )^{2}+\left (3 a \left (y-\frac {a}{3 b x}\right ) x +9\right ) b -2 a^{2}\right ) \sqrt {a^{2}-4 b}+2 a \,\operatorname {arctanh}\left (\frac {6 \left (y-\frac {a}{3 b x}\right ) b x +a}{3 \sqrt {a^{2}-4 b}}\right )}{2 \sqrt {a^{2}-4 b}} = \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 \sqrt {a^{2}-4 b}\, a \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{\textit {\_Z}}+a}{\sqrt {a^{2}-4 b}}\right )-\ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) a^{2}+2 c_{1} a^{2}+2 \textit {\_Z} \,a^{2}+4 \ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) b -8 c_{1} b -8 \textit {\_Z} b \right )}}{x} \]

Problem 3349

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

Problem 3350

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

Problem 3351

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

Problem 3352

ODE	\[ \boxed {y^{\prime }-\operatorname {f1} \left (x \right ) y-\operatorname {f2} \left (x \right ) y^{2}-\operatorname {f3} \left (x \right ) y^{3}=\operatorname {f0} \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 3353

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

Problem 3354

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

Problem 3355

ODE	\[ \boxed {y^{\prime }-g \left (x \right ) y-h \left (x \right ) y^{n}=f \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 3356

ODE	\[ \boxed {y^{\prime }-\sqrt {{| y|}}=0} \]
program solution	\[ \left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0
Maple solution	\[ x +2 \left (\left \{\begin {array}{cc} \sqrt {-y \left (x \right )} & y \left (x \right )\le 0 \\ -\sqrt {y \left (x \right )} & 0

Problem 3357

ODE	\[ \boxed {y^{\prime }-y b -\sqrt {\operatorname {A0} +\operatorname {B0} y}=a} \]
program solution	\[ \int _{}^{y}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple solution	\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{a +b \textit {\_a} +\sqrt {\operatorname {B0} \textit {\_a} +\operatorname {A0}}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 3358

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

Problem 3359

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

Problem 3360

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

Problem 3361

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

Problem 3362

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

Problem 3363

ODE	\[ \boxed {y^{\prime }+\left (f \left (x \right )-y\right ) g \left (x \right ) \sqrt {\left (y-a \right ) \left (y-b \right )}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 3364

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

Problem 3365

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

Problem 3366

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

Problem 3367

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

Problem 3368

ODE	\[ \boxed {y^{\prime }+g \left (x \right ) \sin \left (a y\right )+h \left (x \right ) \cos \left (a y\right )=-f \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 3369

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

Problem 3370

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

Problem 3371

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

Problem 3372

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

Problem 3373

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

Problem 3374

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

Problem 3375

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

Problem 3376

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

Problem 3377

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

Problem 3378

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

Problem 3379

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

Problem 3380

ODE	\[ \boxed {y^{\prime }-\sec \left (x \right )^{2} \sec \left (y\right )^{3}=0} \]
program solution	\[ -\tan \left (x \right )+\frac {\left (2+\cos \left (y\right )^{2}\right ) \sin \left (y\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arctan \left (\frac {3 c_{1} +3 \tan \left (x \right )}{\operatorname {RootOf}\left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 c_{1}^{2}+18 c_{1} \tan \left (x \right )+9 \tan \left (x \right )^{2}-4\right )^{2}+2}, \operatorname {RootOf}\left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 c_{1}^{2}+18 c_{1} \tan \left (x \right )+9 \tan \left (x \right )^{2}-4\right )\right ) \]

Problem 3381

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

Problem 3382

ODE	\[ \boxed {y^{\prime }-\left (\cos \left (x \right ) \sin \left (y\right )+1\right ) \tan \left (y\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 3383

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

Problem 3384

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

Problem 3385

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

Problem 3386

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

Problem 3387

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

Problem 3388

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

Problem 3389

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

Problem 3390

ODE	\[ \boxed {y^{\prime }-x^{m -1} y^{-n +1} f \left (a \,x^{m}+b y^{n}\right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\left (-\frac {-\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} b n \textit {\_a} -\left (m^{\frac {1}{m}}\right )^{m} f \left (a \left (m^{\frac {1}{m}}\right )^{m}+b \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{n}\right ) \left (\left (\frac {b \textit {\_a} -a m}{b}\right )^{\frac {1}{n}}\right )^{-n} a m n +a \,m^{2}}d \textit {\_a} \right ) b \,m^{2}+c_{1} m -x^{m}\right ) b +a \,x^{m}}{b}\right )}^{\frac {1}{n}} \]

Problem 3391

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

Problem 3392

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

Problem 3393

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

Problem 3394

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

Problem 3395

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

Problem 3396

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

Problem 3397

ODE	\[ \boxed {3 y^{\prime }-\sqrt {x^{2}-3 y}=x} \]
program solution	\[ \frac {\ln \left (-x^{2}+4 y\right )}{6}+\frac {\ln \left (y\right )}{3}-\frac {\ln \left (\sqrt {x^{2}-3 y}-x \right )}{3}+\frac {\ln \left (x +\sqrt {x^{2}-3 y}\right )}{3}+\frac {\ln \left (-x +2 \sqrt {x^{2}-3 y}\right )}{6}-\frac {\ln \left (x +2 \sqrt {x^{2}-3 y}\right )}{6} = c_{1} \] Verified OK.
Maple solution	\[ \frac {2 \left (x^{2}-3 y \left (x \right )\right )^{\frac {3}{2}} \left (c_{1} y \left (x \right )^{2} x^{2}-4 c_{1} y \left (x \right )^{3}+1\right )+2 \left (c_{1} y \left (x \right )^{2} x^{2}-4 c_{1} y \left (x \right )^{3}-1\right ) \left (x^{2}-\frac {9 y \left (x \right )}{2}\right ) x}{\left (x^{2}-4 y \left (x \right )\right ) y \left (x \right )^{2} \left (x +\sqrt {x^{2}-3 y \left (x \right )}\right )^{2} \left (-2 \sqrt {x^{2}-3 y \left (x \right )}+x \right )} = 0 \]

Problem 3398

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

Problem 3399

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

Problem 3400

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