2.17.27 Problems 2601 to 2700

Problem 2601

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

Problem 2602

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

Problem 2603

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

Problem 2604

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

Problem 2605

ODE	\[ \boxed {x^{2} y^{\prime \prime }+5 y^{\prime } x +4 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2} \ln \left (x \right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \ln \left (x \right )+c_{1}}{x^{2}} \]

Problem 2606

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

Problem 2607

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

Problem 2608

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

Problem 2609

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

Problem 2610

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

Problem 2611

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

Problem 2612

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

Problem 2613

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

Problem 2614

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

Problem 2615

ODE	\[ \boxed {y^{\prime }=x^{2} \ln \left (x \right )} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ y = \frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}+\frac {19}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}+\frac {19}{9} \]

Problem 2616

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

Problem 2617

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

Problem 2618

ODE	\[ \boxed {y^{\prime \prime }=x \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = \left (-2+x \right ) {\mathrm e}^{x}+5 x +5 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-2+x \right ) {\mathrm e}^{x}+5 x +5 \]

Problem 2619

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

Problem 2620

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x -8 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2} x^{4}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{6}+c_{1}}{x^{2}} \]

Problem 2621

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

Problem 2622

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

Problem 2623

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

Problem 2624

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

Problem 2625

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

Problem 2626

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

Problem 2627

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

Problem 2628

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

Problem 2629

ODE	\[ \boxed {y^{\prime }-\frac {\cos \left (x -y\right )}{\sin \left (x \right ) \sin \left (y\right )}=-1} \]
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 2630

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

Problem 2631

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

Problem 2632

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

Problem 2633

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

Problem 2634

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

Problem 2635

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

Problem 2636

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

Problem 2637

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

Problem 2638

ODE	\[ \boxed {m v^{\prime }+k v^{2}=m g} \] With initial conditions \begin {align*} [v \left (0\right ) = 0] \end {align*}
program solution	\[ v = \munderset {c_{1} \rightarrow 0}{\operatorname {lim}}\frac {\tanh \left (\frac {\sqrt {m g k}\, \left (t +c_{1} \right )}{m}\right ) \sqrt {m g k}}{k} \] Verified OK.
Maple solution	\[ v \left (t \right ) = \frac {\tanh \left (\frac {\sqrt {m g k}\, t}{m}\right ) \sqrt {m g k}}{k} \]

Problem 2639

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

Problem 2640

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

Problem 2641

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

Problem 2642

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

Problem 2643

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

Problem 2644

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

Problem 2645

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

Problem 2646

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

Problem 2647

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

Problem 2648

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

Problem 2649

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

Problem 2650

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

Problem 2651

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

Problem 2652

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

Problem 2653

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

Problem 2654

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

Problem 2655

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

Problem 2656

ODE	\[ \boxed {x^{\prime }+\frac {2 x}{4-t}=5} \] With initial conditions \begin {align*} [x \left (0\right ) = 4] \end {align*}
program solution	\[ x = -\left (t +1\right ) \left (-4+t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -t^{2}+3 t +4 \]

Problem 2657

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

Problem 2658

ODE	\[ \boxed {y^{\prime }-2 y=\left \{\begin {array}{cc} 1 & x \le 1 \\ 0 & 1
program solution	N/A
Maple solution	\[ y \left (x \right ) = \frac {7 \,{\mathrm e}^{2 x}}{2}-\frac {\left (\left \{\begin {array}{cc} 1 & x <1 \\ {\mathrm e}^{2 x -2} & 1\le x \end {array}\right .\right )}{2} \]

Problem 2659

ODE	\[ \boxed {y^{\prime }-2 y=\left \{\begin {array}{cc} 1-x & x <1 \\ 0 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \frac {5 \,{\mathrm e}^{2 x}}{4}+\frac {\left (\left \{\begin {array}{cc} 2 x -1 & x <1 \\ {\mathrm e}^{2 x -2} & 1\le x \end {array}\right .\right )}{4} \]

Problem 2660

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

Problem 2661

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

Problem 2662

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

Problem 2663

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

Problem 2664

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

Problem 2665

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

Problem 2666

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

Problem 2667

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

Problem 2668

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

Problem 2669

ODE	\[ \boxed {y^{\prime } x -\sqrt {16 x^{2}-y^{2}}-y=0} \]
program solution	\[ \arcsin \left (\frac {y}{4 x}\right )-\ln \left (x \right )-c_{2} = 0 \] Verified OK. {0 < x}
Maple solution	\[ -\arctan \left (\frac {y \left (x \right )}{\sqrt {16 x^{2}-y \left (x \right )^{2}}}\right )+\ln \left (x \right )-c_{1} = 0 \]

Problem 2670

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

Problem 2671

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

Problem 2672

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

Problem 2673

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

Problem 2674

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

Problem 2675

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

Problem 2676

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

Problem 2677

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

Problem 2678

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

Problem 2679

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

Problem 2680

ODE	\[ \boxed {y^{\prime }-\frac {2 \left (2 y-x \right )}{y+x}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ -\frac {\left (2 x -y\right )^{3}}{\left (x -y\right )^{2}} = 2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (3 \sqrt {3}\, x \sqrt {x \left (27 x +8\right )}+27 x^{2}+36 x +8\right )^{\frac {1}{3}}}{3}+\frac {4 x +\frac {4}{3}}{\left (3 \sqrt {3}\, x \sqrt {x \left (27 x +8\right )}+27 x^{2}+36 x +8\right )^{\frac {1}{3}}}+2 x +\frac {2}{3} \]

Problem 2681

ODE	\[ \boxed {y^{\prime }-\frac {2 x -y}{x +4 y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ -\frac {\left (y+x \right ) \left (-2 y+x \right )}{2} = 1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{4}+\frac {\sqrt {9 x^{2}+16}}{4} \]

Problem 2682

ODE	\[ \boxed {y^{\prime }-\frac {y-\sqrt {x^{2}+y^{2}}}{x}=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 4] \end {align*}
program solution	\[ \operatorname {arcsinh}\left (\frac {y}{x}\right )+\ln \left (x \right )-\operatorname {arcsinh}\left (\frac {4}{3}\right )-\ln \left (3\right ) = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}}{2}-\frac {1}{2} \\ y \left (x \right ) &= -\frac {x^{2}}{18}+\frac {9}{2} \\ \end{align*}

Problem 2683

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

Problem 2684

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

Problem 2685

ODE	\[ \boxed {y^{\prime }-\frac {x +\frac {y}{2}}{\frac {x}{2}-y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ \ln \left (\frac {y^{2}}{x^{2}}+1\right )-\arctan \left (\frac {y}{x}\right )+2 \ln \left (x \right )-\ln \left (2\right )+\frac {\pi }{4} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (4 \textit {\_Z} -4 \ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )-8 \ln \left (x \right )+4 \ln \left (2\right )-\pi \right )\right ) x \]

Problem 2686

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

Problem 2687

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

Problem 2688

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

Problem 2689

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

Problem 2690

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

Problem 2691

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

Problem 2692

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

Problem 2693

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

Problem 2694

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

Problem 2695

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

Problem 2696

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

Problem 2697

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

Problem 2698

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

Problem 2699

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

Problem 2700

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