2.17.125 Problems 12401 to 12500

Problem 12401

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (36 x^{5}+25 x^{4}+16 x^{3}+9 x^{2}+4 x +1+O\left (x^{6}\right )\right )+c_{2} \left (\left (36 x^{5}+25 x^{4}+16 x^{3}+9 x^{2}+4 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )-60 x^{5}-40 x^{4}-24 x^{3}-12 x^{2}-4 x +O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+4 x +9 x^{2}+16 x^{3}+25 x^{4}+36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -12 x^{2}-24 x^{3}-40 x^{4}-60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 12402

ODE	\[ \boxed {\left (x^{2}-1\right )^{2} y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{6} x^{4}-\frac {7}{60} x^{5}+\frac {37}{360} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}+\frac {7}{60} x^{5}-\frac {37}{360} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{6} x^{4}-\frac {7}{60} x^{5}\right ) c_{1} +\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}+\frac {7}{60} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{6} x^{4}-\frac {7}{60} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{6} x^{4}+\frac {7}{60} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 12403

ODE	\[ \boxed {x y^{\prime \prime }+4 y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{280}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}-\frac {x^{4}}{8}+O\left (x^{6}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

Problem 12404

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }-k y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\left (\frac {k}{3}-\frac {1}{6}\right ) x +\left (\frac {1}{30} k^{2}-\frac {1}{15} k +\frac {1}{40}\right ) x^{2}+\frac {\left (2 k -5\right ) \left (2 k -3\right ) \left (2 k -1\right ) x^{3}}{5040}+\frac {\left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (2 k -1\right ) x^{4}}{362880}+\frac {\left (2 k -9\right ) \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (2 k -1\right ) x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} \left (1+k x +\frac {\left (k -1\right ) k \,x^{2}}{6}+\frac {\left (k -2\right ) \left (k -1\right ) k \,x^{3}}{90}+\frac {\left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k \,x^{4}}{2520}+\frac {\left (k -4\right ) \left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k \,x^{5}}{113400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, c_{1} \left (1+\left (\frac {k}{3}-\frac {1}{6}\right ) x +\left (\frac {1}{30} k^{2}-\frac {1}{15} k +\frac {1}{40}\right ) x^{2}+\frac {1}{5040} \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{3}+\frac {1}{362880} \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{4}+\frac {1}{39916800} \left (2 k -9\right ) \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+k x +\frac {1}{6} \left (-1+k \right ) k x^{2}+\frac {1}{90} \left (-2+k \right ) \left (-1+k \right ) k x^{3}+\frac {1}{2520} \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{4}+\frac {1}{113400} \left (-4+k \right ) \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 12405

ODE	\[ \boxed {x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 12406

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 12407

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{5}+\frac {x^{2}}{35}+\frac {x^{3}}{315}+\frac {x^{4}}{3465}+\frac {x^{5}}{45045}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 12408

ODE	\[ \boxed {x \left (x -1\right ) y^{\prime \prime }+3 y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-x^{2}-2 x^{3}-3 x^{4}-4 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{2} +\left (1+3 x +5 x^{2}+7 x^{3}+9 x^{4}+11 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 12409

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

Problem 12410

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

Problem 12411

ODE	\[ \boxed {x y^{\prime \prime }+\left (1+x \right )^{2} y=0} \]
program solution	\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 12412

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

Problem 12413

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

Problem 12414

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

Problem 12415

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

Problem 12416

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\nu ^{2}+x^{2}\right ) y=\sin \left (x \right )} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (\nu , x\right )+c_{2} \operatorname {BesselY}\left (\nu , x\right )-\frac {\left (\operatorname {BesselJ}\left (\nu , x\right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}-\frac {\nu }{2}, \frac {3}{4}-\frac {\nu }{2}, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}, 1-\nu , \frac {3}{2}-\nu , \frac {3}{2}-\frac {\nu }{2}\right ], -x^{2}\right ) \Gamma \left (\nu +2\right )^{2} 2^{\nu } x^{-\nu }+\operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\nu }{2}, \frac {3}{4}+\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {3}{2}, \nu +1, \frac {3}{2}+\nu , \frac {3}{2}+\frac {\nu }{2}\right ], -x^{2}\right ) \pi \left (\operatorname {BesselJ}\left (\nu , x\right ) \cot \left (\pi \nu \right )-\operatorname {BesselY}\left (\nu , x\right )\right ) 2^{-\nu } x^{\nu } \left (\nu -1\right ) \nu \left (\nu +1\right )\right ) x}{2 \left (\nu -1\right ) \nu \left (\nu +1\right ) \Gamma \left (\nu +2\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x^{1-\nu } 2^{\nu -1} \operatorname {BesselJ}\left (\nu , x\right ) \Gamma \left (\nu +2\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\nu }{2}, \frac {5}{4}-\frac {\nu }{2}, \frac {3}{4}-\frac {\nu }{2}\right ], \left [\frac {3}{2}, 1-\nu , \frac {3}{2}-\nu , \frac {3}{2}-\frac {\nu }{2}\right ], -x^{2}\right )}{\nu \left (\nu -1\right ) \left (\nu +1\right )}+\operatorname {BesselJ}\left (\nu , x\right ) c_{2} +\operatorname {BesselY}\left (\nu , x\right ) c_{1} -\frac {\pi 2^{-1-\nu } x^{\nu +1} \left (\operatorname {BesselJ}\left (\nu , x\right ) \cot \left (\pi \nu \right )-\operatorname {BesselY}\left (\nu , x\right )\right ) \operatorname {hypergeom}\left (\left [\frac {\nu }{2}+\frac {1}{2}, \frac {5}{4}+\frac {\nu }{2}, \frac {3}{4}+\frac {\nu }{2}\right ], \left [\frac {3}{2}, \nu +1, \frac {3}{2}+\nu , \frac {3}{2}+\frac {\nu }{2}\right ], -x^{2}\right )}{\Gamma \left (\nu +2\right )} \]

Problem 12417

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

Problem 12418

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

Problem 12419

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

Problem 12420

ODE	\[ \boxed {x y \left (1-{y^{\prime }}^{2}\right )-\left (x^{2}-y^{2}-a^{2}\right ) y^{\prime }=0} \]
program solution	\[ \frac {-\ln \left (\left (a^{2}-x^{2}\right ) \sqrt {\left (y^{2}+\left (a +x \right )^{2}\right ) \left (y^{2}+\left (a -x \right )^{2}\right )}+a^{4}+\left (-2 x^{2}+y^{2}\right ) a^{2}+x^{4}+y^{2} x^{2}\right )+4 \ln \left (y\right )-\ln \left (a^{2}+x^{2}+y^{2}+\sqrt {\left (y^{2}+\left (a +x \right )^{2}\right ) \left (y^{2}+\left (a -x \right )^{2}\right )}\right )-\ln \left (2\right )}{4 a^{2}} = c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 12421

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

Problem 12422

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

Problem 12423

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x -a^{2} y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-1\right )^{\frac {1}{4}} {\mathrm e}^{-a \arcsin \left (x \right )}}{\left (x -1\right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}}+\frac {c_{2} \left (x^{2}-1\right )^{\frac {1}{4}} {\mathrm e}^{-a \arcsin \left (x \right )} \left (\int \frac {{\mathrm e}^{2 a \arcsin \left (x \right )}}{\sqrt {x^{2}-1}}d x \right )}{\left (x -1\right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x +\sqrt {x^{2}-1}\right )^{i a}+c_{2} \left (x +\sqrt {x^{2}-1}\right )^{-i a} \]

Problem 12424

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

Problem 12425

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

Problem 12426

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

Problem 12427

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

Problem 12428

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

Problem 12429

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

Problem 12430

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

Problem 12431

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

Problem 12432

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

Problem 12433

ODE	\[ \boxed {r^{\prime }+r \tan \left (t \right )=0} \]
program solution	\[ r = {\mathrm e}^{-c_{1}} \cos \left (t \right ) \] Verified OK.
Maple solution	\[ r \left (t \right ) = \cos \left (t \right ) c_{1} \]

Problem 12434

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

Problem 12435

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

Problem 12436

ODE	\[ \boxed {3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime }=0} \]
program solution	\[ -\ln \left ({\mathrm e}^{x}-1\right )+\frac {\ln \left (\tan \left (y\right )\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\arctan \left (-\frac {2 c_{1} \left ({\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-1\right )}{-c_{1}^{2} {\mathrm e}^{6 x}+6 c_{1}^{2} {\mathrm e}^{5 x}-15 c_{1}^{2} {\mathrm e}^{4 x}+20 c_{1}^{2} {\mathrm e}^{3 x}-15 c_{1}^{2} {\mathrm e}^{2 x}+6 c_{1}^{2} {\mathrm e}^{x}-c_{1}^{2}-1}, \frac {c_{1}^{2} {\mathrm e}^{6 x}-6 c_{1}^{2} {\mathrm e}^{5 x}+15 c_{1}^{2} {\mathrm e}^{4 x}-20 c_{1}^{2} {\mathrm e}^{3 x}+15 c_{1}^{2} {\mathrm e}^{2 x}-6 c_{1}^{2} {\mathrm e}^{x}+c_{1}^{2}-1}{-c_{1}^{2} {\mathrm e}^{6 x}+6 c_{1}^{2} {\mathrm e}^{5 x}-15 c_{1}^{2} {\mathrm e}^{4 x}+20 c_{1}^{2} {\mathrm e}^{3 x}-15 c_{1}^{2} {\mathrm e}^{2 x}+6 c_{1}^{2} {\mathrm e}^{x}-c_{1}^{2}-1}\right )}{2} \]

Problem 12437

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

Problem 12438

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

Problem 12439

ODE	\[ \boxed {y^{\prime } x +y=-x} \]
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 12440

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

Problem 12441

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

Problem 12442

ODE	\[ \boxed {8 y+\left (5 y+7 x \right ) y^{\prime }=-10 x} \]
program solution	\[ \frac {3 \ln \left (y+2 x \right )}{5}+\frac {2 \ln \left (x +y\right )}{5} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (-2+\operatorname {RootOf}\left (\textit {\_Z}^{25} c_{1} x^{5}-2 \textit {\_Z}^{20} c_{1} x^{5}+\textit {\_Z}^{15} c_{1} x^{5}-1\right )^{5}\right ) \]

Problem 12443

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

Problem 12444

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

Problem 12445

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

Problem 12446

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

Problem 12447

ODE	\[ \boxed {3 y-\left (3 x -7 y-3\right ) y^{\prime }=7 x -7} \]
program solution	\[ \frac {5 \ln \left (x -1+y\right )}{3}+\frac {2 \ln \left (-x +1+y\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 12448

ODE	\[ \boxed {2 y-\left (2 x +4 y+3\right ) y^{\prime }=-1-x} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left ({\mathrm e}^{8 x +5+16 c_{1}}\right )}{8}-\frac {x}{2}-\frac {5}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2}+\frac {\operatorname {LambertW}\left (c_{1} {\mathrm e}^{5+8 x}\right )}{8}-\frac {5}{8} \]

Problem 12449

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

Problem 12450

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

Problem 12451

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

Problem 12452

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

Problem 12453

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

Problem 12454

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

Problem 12455

ODE	\[ \boxed {y^{\prime }-\frac {a y}{x}=\frac {1+x}{x}} \]
program solution	\[ y = -\frac {\left (x^{-a} a x -c_{1} a^{2}+x^{-a} a +c_{1} a -x^{-a}\right ) x^{a}}{a \left (a -1\right )} \] 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 12456

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

Problem 12457

ODE	\[ \boxed {s^{\prime } \cos \left (t \right )+s \sin \left (t \right )=1} \]
program solution	\[ s = \frac {\tan \left (t \right )+c_{1}}{\sec \left (t \right )} \] Verified OK.
Maple solution	\[ s \left (t \right ) = \cos \left (t \right ) c_{1} +\sin \left (t \right ) \]

Problem 12458

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

Problem 12459

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

Problem 12460

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

Problem 12461

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

Problem 12462

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

Problem 12463

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

Problem 12464

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

Problem 12465

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

Problem 12466

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

Problem 12467

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

Problem 12468

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

Problem 12469

ODE	\[ \boxed {y+\left (-2 y+x \right ) y^{\prime }=-x^{2}} \]
program solution	\[ \frac {x^{3}}{3}+y x -y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x}{2}-\frac {\sqrt {12 x^{3}+9 x^{2}+36 c_{1}}}{6} \\ y \left (x \right ) &= \frac {x}{2}+\frac {\sqrt {12 x^{3}+9 x^{2}+36 c_{1}}}{6} \\ \end{align*}

Problem 12470

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

Problem 12471

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

Problem 12472

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

Problem 12473

ODE	\[ \boxed {6 x y^{2}+3 \left (2 y x^{2}+y^{2}\right ) y^{\prime }=-4 x^{3}} \]
program solution	\[ \frac {\left (2 x^{2}+3 y^{2}\right )^{2}}{4}-\frac {9 y^{4}}{4}+y^{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {4 x^{10}+x^{8}+4 c_{1} x^{6}+2 c_{1} x^{4}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{4}}{\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {4 x^{10}+x^{8}+4 c_{1} x^{6}+2 c_{1} x^{4}+c_{1}^{2}}\right )^{\frac {1}{3}}}-x^{2} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {2}{3}}-4 x^{4}-4 x^{2} \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}-\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {2}{3}}}{4 \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}\right ) x^{2}}{\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 12474

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

Problem 12475

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

Problem 12476

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

Problem 12477

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

Problem 12478

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

Problem 12479

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

Problem 12480

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

Problem 12481

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

Problem 12482

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

Problem 12483

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

Problem 12484

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

Problem 12485

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

Problem 12486

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

Problem 12487

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

Problem 12488

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

Problem 12489

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

Problem 12490

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

Problem 12491

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

Problem 12492

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

Problem 12493

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

Problem 12494

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

Problem 12495

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

Problem 12496

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

Problem 12497

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

Problem 12498

ODE
program solution
Maple solution	\[ y \left (x \right ) = \left (-c_{1} -x \right ) \ln \left (c_{1} +x \right )+\left (c_{2} +1\right ) x +c_{3} +c_{1} \]

Problem 12499

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

Problem 12500

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