2.17.58 Problems 5701 to 5800

Problem 5701

ODE	\[ \boxed {y^{\prime \prime }+y=\left \{\begin {array}{cc} t & 0
program solution	\[ y = -\sin \left (t \right )+\left (\left \{\begin {array}{cc} t & t <1 \\ \cos \left (t -1\right )+\sin \left (t -1\right ) & 1\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\sin \left (t \right )+\left (\left \{\begin {array}{cc} t & t <1 \\ \sin \left (t -1\right )+\cos \left (t -1\right ) & 1\le t \end {array}\right .\right ) \]

Problem 5702

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=\left \{\begin {array}{cc} 10 \sin \left (t \right ) & 0
program solution	\[ y = \left \{\begin {array}{cc} 2 \sin \left (t \right ) \cos \left (t \right ) {\mathrm e}^{-t}-\cos \left (t \right )+2 \sin \left (t \right ) & t <2 \pi \\ \frac {\left (-2 \cos \left (2 t \right )+\sin \left (2 t \right )\right ) {\mathrm e}^{2 \pi -t}}{2}+\sin \left (2 t \right ) {\mathrm e}^{-t} & 2 \pi \le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} \sin \left (2 t \right ) {\mathrm e}^{-t}-\cos \left (t \right )+2 \sin \left (t \right ) & t <2 \pi \\ -2 & t =2 \pi \\ \sin \left (2 t \right ) {\mathrm e}^{-t}+\frac {\left (-2 \cos \left (2 t \right )+\sin \left (2 t \right )\right ) {\mathrm e}^{2 \pi -t}}{2} & 2 \pi

Problem 5703

ODE	\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 8 t^{2} & 0
program solution	\[ y = \cos \left (2 t \right )+\left (\left \{\begin {array}{cc} 2 t^{2}-1 & t <5 \\ 49 \cos \left (-10+2 t \right )+10 \sin \left (-10+2 t \right ) & 5\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (2 t \right )+\left (\left \{\begin {array}{cc} 2 t^{2}-1 & t <5 \\ 10 \sin \left (2 t -10\right )+49 \cos \left (2 t -10\right ) & 5\le t \end {array}\right .\right ) \]

Problem 5704

ODE	\[ \boxed {y^{\prime \prime }+4 y=\delta \left (t -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 8, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 8 \cos \left (2 t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ \frac {\sin \left (2 t \right )}{2} & \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -\pi \right ) \sin \left (2 t \right )}{2}+8 \cos \left (2 t \right ) \]

Problem 5705

ODE	\[ \boxed {y^{\prime \prime }+16 y=4 \delta \left (t -3 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 2 \cos \left (4 t \right )+\left (\left \{\begin {array}{cc} 0 & t <3 \pi \\ \sin \left (4 t \right ) & 3 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -3 \pi \right ) \sin \left (4 t \right )+2 \cos \left (4 t \right ) \]

Problem 5706

ODE	\[ \boxed {y^{\prime \prime }+y=\delta \left (t -\pi \right )-\delta \left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \sin \left (t \right ) \left (\left \{\begin {array}{cc} 1 & t <\pi \\ 0 & t \le 2 \pi \\ -1 & 2 \pi
Maple solution	\[ y \left (t \right ) = \sin \left (t \right ) \left (1-\operatorname {Heaviside}\left (t -2 \pi \right )-\operatorname {Heaviside}\left (t -\pi \right )\right ) \]

Problem 5707

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+5 y=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = 3 \,{\mathrm e}^{-2 t} \sin \left (t \right )+\left (\left \{\begin {array}{cc} 0 & t <1 \\ {\mathrm e}^{-2 t +2} \sin \left (t -1\right ) & 1\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2} \sin \left (t -1\right )+3 \,{\mathrm e}^{-2 t} \sin \left (t \right ) \]

Problem 5708

ODE	\[ \boxed {4 y^{\prime \prime }+24 y^{\prime }+37 y=17 \,{\mathrm e}^{-t}+\delta \left (t -\frac {1}{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{-t}+4 \,{\mathrm e}^{-3 t} \sin \left (\frac {t}{2}\right )+\left (\left \{\begin {array}{cc} 0 & t <\frac {1}{2} \\ \frac {{\mathrm e}^{-3 t +\frac {3}{2}} \sin \left (\frac {t}{2}-\frac {1}{4}\right )}{2} & \frac {1}{2}\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -\frac {1}{2}\right ) {\mathrm e}^{-3 t +\frac {3}{2}} \sin \left (-\frac {1}{4}+\frac {t}{2}\right )}{2}+4 \,{\mathrm e}^{-3 t} \sin \left (\frac {t}{2}\right )+{\mathrm e}^{-t} \]

Problem 5709

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=10 \sin \left (t \right )+10 \delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \sin \left (t \right )-3 \cos \left (t \right )+6 \,{\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t}+\left (\left \{\begin {array}{cc} 0 & t <1 \\ 10 \,{\mathrm e}^{-t +1}-10 \,{\mathrm e}^{-2 t +2} & 1\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -10 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}+10 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}-2 \,{\mathrm e}^{-2 t}+\sin \left (t \right )-3 \cos \left (t \right )+6 \,{\mathrm e}^{-t} \]

Problem 5710

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+5 y=\left (1-\operatorname {Heaviside}\left (-10+t \right )\right ) {\mathrm e}^{t}-{\mathrm e}^{10} \delta \left (-10+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\frac {\left (\left \{\begin {array}{cc} \left (-{\mathrm e}^{3 t}+\cos \left (t \right )-7 \sin \left (t \right )\right ) {\mathrm e}^{-2 t} & t <10 \\ \left (-2 \,{\mathrm e}^{30}+\cos \left (10\right )-7 \sin \left (10\right )\right ) {\mathrm e}^{-20} & t =10 \\ \left (-\cos \left (-10+t \right )+7 \sin \left (-10+t \right )\right ) {\mathrm e}^{30-2 t}+\left (\cos \left (t \right )-7 \sin \left (t \right )\right ) {\mathrm e}^{-2 t} & 10
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (\left (-{\mathrm e}^{3 t}+\left (\left (-7 \cos \left (10\right )+\sin \left (10\right )\right ) \sin \left (t \right )+\left (\cos \left (10\right )+7 \sin \left (10\right )\right ) \cos \left (t \right )\right ) {\mathrm e}^{30}\right ) \operatorname {Heaviside}\left (t -10\right )-\cos \left (t \right )+7 \sin \left (t \right )+{\mathrm e}^{3 t}\right )}{10} \]

Problem 5711

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=\delta \left (t -\frac {\pi }{2}\right )+\operatorname {Heaviside}\left (t -\pi \right ) \cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <\frac {\pi }{2} \\ {\mathrm e}^{-2 t +\pi }-{\mathrm e}^{-3 t +\frac {3 \pi }{2}} & t <\pi \\ \frac {2 \,{\mathrm e}^{2 \pi -2 t}}{5}-\frac {3 \,{\mathrm e}^{3 \pi -3 t}}{10}+\frac {\sin \left (t \right )}{10}+\frac {\cos \left (t \right )}{10}+{\mathrm e}^{-2 t +\pi }-{\mathrm e}^{-3 t +\frac {3 \pi }{2}} & \pi \le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-3 t +\frac {3 \pi }{2}}-\frac {3 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-3 t +3 \pi }}{10}+\frac {2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-2 t +2 \pi }}{5}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-2 t +\pi }+\frac {\operatorname {Heaviside}\left (t -\pi \right ) \left (\cos \left (t \right )+\sin \left (t \right )\right )}{10} \]

Problem 5712

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y=\operatorname {Heaviside}\left (t -1\right )+\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{-2 t}-{\mathrm e}^{-3 t}+\frac {\left (\left \{\begin {array}{cc} 0 & t <1 \\ 1+2 \,{\mathrm e}^{-3 t +3}-3 \,{\mathrm e}^{-2 t +2} & t <2 \\ 6 \,{\mathrm e}^{-2 t +4}-6 \,{\mathrm e}^{6-3 t}+1+2 \,{\mathrm e}^{-3 t +3}-3 \,{\mathrm e}^{-2 t +2} & 2\le t \end {array}\right .\right )}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -{\mathrm e}^{-3 t}+{\mathrm e}^{-2 t}+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2 t +4}-\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2}}{2}+\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{3}+\frac {\operatorname {Heaviside}\left (t -1\right )}{6} \]

Problem 5713

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=25 t -100 \delta \left (t -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = -2+5 t -\left (\left \{\begin {array}{cc} 0 & t <\pi \\ 50 \,{\mathrm e}^{\pi -t} \sin \left (2 t \right ) & \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -50 \operatorname {Heaviside}\left (t -\pi \right ) \sin \left (2 t \right ) {\mathrm e}^{\pi -t}+5 t -2 \]

Problem 5714

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{y}=0} \]
program solution	\[ y = \frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \] Verified OK. \[ y = -\frac {\sqrt {6 x^{3}+18 c_{1}}}{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 x^{3}+9 c_{1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6 x^{3}+9 c_{1}}}{3} \\ \end{align*}

Problem 5715

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{y \left (x^{3}+1\right )}=0} \]
program solution	\[ y = \frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \] Verified OK. \[ y = -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+18 c_{1}}}{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 \ln \left (x^{3}+1\right )+9 c_{1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6 \ln \left (x^{3}+1\right )+9 c_{1}}}{3} \\ \end{align*}

Problem 5716

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

Problem 5717

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

Problem 5718

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{1+y^{2}}=0} \]
program solution	\[ y = \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \] Verified OK. \[ y = -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK. \[ y = -\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}-4}{2 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}-4}{4 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}} \sqrt {3}+4 i \sqrt {3}-\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {2}{3}}+4}{4 \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 5719

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

Problem 5720

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

Problem 5721

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

Problem 5722

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

Problem 5723

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

Problem 5724

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

Problem 5725

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

Problem 5726

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

Problem 5727

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

Problem 5728

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

Problem 5729

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

Problem 5730

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

Problem 5731

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

Problem 5732

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

Problem 5733

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

Problem 5734

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

Problem 5735

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

Problem 5736

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

Problem 5737

ODE	\[ \boxed {z^{\prime }-10^{x +z}=0} \]
program solution	\[ -\frac {10^{-z}}{\ln \left (10\right )}-\frac {10^{x}}{\ln \left (10\right )}-c_{1} = 0 \] Verified OK.
Maple solution	\[ z \left (x \right ) = \frac {\ln \left (-\frac {1}{c_{1} \ln \left (2\right )+c_{1} \ln \left (5\right )+10^{x}}\right )}{\ln \left (2\right )+\ln \left (5\right )} \]

Problem 5738

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

Problem 5739

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

Problem 5740

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

Problem 5741

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

Problem 5742

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

Problem 5743

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

Problem 5744

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

Problem 5745

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

Problem 5746

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

Problem 5747

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

Problem 5748

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

Problem 5749

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

Problem 5750

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

Problem 5751

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

Problem 5752

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

Problem 5753

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

Problem 5754

ODE	\[ \boxed {y^{\prime } x -y-\tan \left (\frac {y}{x}\right ) x=0} \]
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 5755

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

Problem 5756

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

Problem 5757

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

Problem 5758

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

Problem 5759

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

Problem 5760

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

Problem 5761

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

Problem 5762

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

Problem 5763

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

Problem 5764

ODE	\[ \boxed {y x +y^{2}-x^{2} y^{\prime }=-x^{2}} \]
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 5765

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

Problem 5766

ODE	\[ \boxed {y^{\prime }-\frac {2 x y}{3 x^{2}-y^{2}}=0} \]
program solution	\[ -\frac {y^{3}}{\left (x -y\right ) \left (x +y\right )} = c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}}{3 c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 5767

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

Problem 5768

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

Problem 5769

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

Problem 5770

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

Problem 5771

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

Problem 5772

ODE	\[ \boxed {\left (y^{\prime } x +y\right )^{2}-y^{\prime } y^{2}=0} \]
program solution	\[ {\left (\frac {\sqrt {\frac {y^{2}-4 y x}{x^{2}}}\, x +y-2 x}{y}\right )}^{\frac {1}{4}} = c_{3} {\mathrm e}^{c_{2}} \sqrt {x} \] Verified OK. {0 < x} \[ {\left (\frac {\sqrt {\frac {y^{2}-4 y x}{x^{2}}}\, x +2 x -y}{y}\right )}^{\frac {1}{4}} = c_{6} \sqrt {x}\, {\mathrm e}^{c_{5}} \] Verified OK. {0 < x}
Maple solution	\begin{align*} y \left (x \right ) &= 4 x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {2 c_{1}^{2} \left (-\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= -\frac {2 c_{1}^{2} \left (\sqrt {2}\, c_{1} +x \right )}{-2 c_{1}^{2}+x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{3} \sqrt {2}-2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ y \left (x \right ) &= \frac {c_{1}^{2} \left (\sqrt {2}\, c_{1} +2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}

Problem 5773

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

Problem 5774

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

Problem 5775

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

Problem 5776

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

Problem 5777

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

Problem 5778

ODE	\[ \boxed {y^{\prime } x -\frac {y}{2}=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = 2 x +\sqrt {x}\, c_{1} \]

Problem 5779

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

Problem 5780

ODE	\[ \boxed {-4 y+\left (x +y-2\right ) y^{\prime }=-2 x -6} \]
program solution	\[ -\frac {\left (-y+2 x +1\right )^{3}}{\left (x +\frac {4}{3}-y\right )^{2}} = c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {2 \left (\left (\frac {i \sqrt {3}}{72}-\frac {1}{72}\right ) \left (36 \sqrt {3}\, \left (x -\frac {1}{3}\right ) c_{1}^{2} \sqrt {\frac {243 \left (x -\frac {1}{3}\right )^{2} c_{1} -12 x +4}{c_{1}}}+8+972 \left (x -\frac {1}{3}\right )^{2} c_{1}^{2}+\left (-216 x +72\right ) c_{1} \right )^{\frac {2}{3}}+\left (\frac {1}{18}+\left (-x -\frac {1}{2}\right ) c_{1} \right ) \left (36 \sqrt {3}\, \left (x -\frac {1}{3}\right ) c_{1}^{2} \sqrt {\frac {243 \left (x -\frac {1}{3}\right )^{2} c_{1} -12 x +4}{c_{1}}}+8+972 \left (x -\frac {1}{3}\right )^{2} c_{1}^{2}+\left (-216 x +72\right ) c_{1} \right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (-\frac {1}{18}+\left (x -\frac {1}{3}\right ) c_{1} \right )\right )}{\left (36 \sqrt {3}\, \left (x -\frac {1}{3}\right ) c_{1}^{2} \sqrt {\frac {243 \left (x -\frac {1}{3}\right )^{2} c_{1} -12 x +4}{c_{1}}}+8+972 \left (x -\frac {1}{3}\right )^{2} c_{1}^{2}+\left (-216 x +72\right ) c_{1} \right )^{\frac {1}{3}} c_{1}} \]

Problem 5781

ODE	\[ \boxed {y^{\prime }-\frac {2 y-x +5}{2 x -y-4}=0} \]
program solution	\[ -\frac {\left (x +y+1\right )^{3}}{-y+x -3} = c_{4} {\mathrm e}^{2 c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (i \sqrt {3}-1\right ) \left (27 c_{1} \left (x -1\right )+3 \sqrt {3}\, \sqrt {27 \left (x -1\right )^{2} c_{1}^{2}-1}\right )^{\frac {2}{3}}-3 i \sqrt {3}-3+6 \left (3 \sqrt {3}\, \sqrt {27 \left (x -1\right )^{2} c_{1}^{2}-1}+27 c_{1} x -27 c_{1} \right )^{\frac {1}{3}} \left (x +1\right ) c_{1}}{6 \left (27 c_{1} \left (x -1\right )+3 \sqrt {3}\, \sqrt {27 \left (x -1\right )^{2} c_{1}^{2}-1}\right )^{\frac {1}{3}} c_{1}} \]

Problem 5782

ODE	\[ \boxed {y^{\prime }+\frac {4 x +3 y+15}{2 x +y+7}=0} \]
program solution	\[ \left (4 x +13+y\right )^{2} \left (y+4+x \right ) = c_{4} {\mathrm e}^{3 c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-24 \left (x +3\right )^{2} c_{1} \left (x +\frac {10}{3}\right ) {\left (4 \sqrt {-4 \left (-\frac {1}{4}+\left (x +3\right )^{3} c_{1} \right ) \left (x +3\right )^{6} c_{1}^{2}}+4 \left (x^{3}+9 x^{2}+27 x +27\right ) c_{1} \right )}^{\frac {2}{3}}+i \left (-16 \left (x +3\right )^{6} c_{1}^{2}+\left (4 c_{1} x^{3}+36 c_{1} x^{2}+108 c_{1} x +4 \sqrt {-4 \left (-\frac {1}{4}+\left (x +3\right )^{3} c_{1} \right ) \left (x +3\right )^{6} c_{1}^{2}}+108 c_{1} \right )^{\frac {4}{3}}\right ) \sqrt {3}+16 \left (x +3\right )^{6} c_{1}^{2}+\left (4 c_{1} x^{3}+36 c_{1} x^{2}+108 c_{1} x +4 \sqrt {-4 \left (-\frac {1}{4}+\left (x +3\right )^{3} c_{1} \right ) \left (x +3\right )^{6} c_{1}^{2}}+108 c_{1} \right )^{\frac {4}{3}}}{8 {\left (4 \sqrt {-4 \left (-\frac {1}{4}+\left (x +3\right )^{3} c_{1} \right ) \left (x +3\right )^{6} c_{1}^{2}}+4 \left (x^{3}+9 x^{2}+27 x +27\right ) c_{1} \right )}^{\frac {2}{3}} \left (x +3\right )^{2} c_{1}} \]

Problem 5783

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

Problem 5784

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

Problem 5785

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

Problem 5786

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

Problem 5787

ODE	\[ \boxed {\left (4 y+x \right ) y^{\prime }-3 y=2 x -5} \]
program solution	\[ y = \left (x -4\right ) \operatorname {RootOf}\left (\left (2 x^{6}-48 x^{5}+480 x^{4}-2560 x^{3}+7680 x^{2}-12288 x +8192\right ) \textit {\_Z}^{6}+\left (-9 x^{6}+216 x^{5}-2160 x^{4}+11520 x^{3}-34560 x^{2}+55296 x -36864\right ) \textit {\_Z}^{5}+\left (15 x^{6}-360 x^{5}+3600 x^{4}-19200 x^{3}+57600 x^{2}-92160 x +61440\right ) \textit {\_Z}^{4}+\left (-10 x^{6}+240 x^{5}-2400 x^{4}+12800 x^{3}-38400 x^{2}+61440 x -40960\right ) \textit {\_Z}^{3}+\left (3 x^{6}-72 x^{5}+720 x^{4}-3840 x^{3}+11520 x^{2}-18432 x +12288\right ) \textit {\_Z} -x^{6}+24 x^{5}-240 x^{4}+1280 x^{3}-2 c_{4} {\mathrm e}^{3 c_{3}}-3840 x^{2}+6144 x -4096\right )-1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x -5\right ) \operatorname {RootOf}\left (\textit {\_Z}^{36}+\left (3 c_{1} x^{6}-72 c_{1} x^{5}+720 c_{1} x^{4}-3840 c_{1} x^{3}+11520 c_{1} x^{2}-18432 c_{1} x +12288 c_{1} \right ) \textit {\_Z}^{6}-2 c_{1} x^{6}+48 c_{1} x^{5}-480 c_{1} x^{4}+2560 c_{1} x^{3}-7680 c_{1} x^{2}+12288 c_{1} x -8192 c_{1} \right )^{6}-x +4}{\operatorname {RootOf}\left (\textit {\_Z}^{36}+\left (3 c_{1} x^{6}-72 c_{1} x^{5}+720 c_{1} x^{4}-3840 c_{1} x^{3}+11520 c_{1} x^{2}-18432 c_{1} x +12288 c_{1} \right ) \textit {\_Z}^{6}-2 c_{1} x^{6}+48 c_{1} x^{5}-480 c_{1} x^{4}+2560 c_{1} x^{3}-7680 c_{1} x^{2}+12288 c_{1} x -8192 c_{1} \right )^{6}} \]

Problem 5788

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

Problem 5789

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

Problem 5790

ODE	\[ \boxed {y^{\prime }-\frac {x -2 y+5}{y-2 x -4}=0} \]
program solution	\[ -\frac {\left (-1+y+x \right )^{3}}{3+x -y} = c_{4} {\mathrm e}^{2 c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\frac {1}{2}+\frac {\left (1-i \sqrt {3}\right ) \left (27 \left (x +1\right ) c_{1} +3 \sqrt {3}\, \sqrt {27 \left (x +1\right )^{2} c_{1}^{2}-1}\right )^{\frac {2}{3}}}{6}+\frac {i \sqrt {3}}{2}-\left (3 \sqrt {3}\, \sqrt {27 \left (x +1\right )^{2} c_{1}^{2}-1}+27 c_{1} x +27 c_{1} \right )^{\frac {1}{3}} \left (x -1\right ) c_{1}}{\left (27 \left (x +1\right ) c_{1} +3 \sqrt {3}\, \sqrt {27 \left (x +1\right )^{2} c_{1}^{2}-1}\right )^{\frac {1}{3}} c_{1}} \]

Problem 5791

ODE	\[ \boxed {y^{\prime }-\frac {3 x -y+1}{2 x +y+4}=0} \]
program solution	\[ \frac {\ln \left (\frac {\left (2+y\right )^{2}}{\left (1+x \right )^{2}}+\frac {6+3 y}{1+x}-3\right )}{2}-\frac {\sqrt {21}\, \operatorname {arctanh}\left (\frac {\left (2 y+7+3 x \right ) \sqrt {21}}{21+21 x}\right )}{21}+\ln \left (1+x \right )-c_{3} = 0 \] Verified OK.
Maple solution	\[ -\frac {\ln \left (\frac {y \left (x \right )^{2}+\left (3 x +7\right ) y \left (x \right )-3 x^{2}+7}{\left (x +1\right )^{2}}\right )}{2}+\frac {\sqrt {21}\, \operatorname {arctanh}\left (\frac {\left (2 y \left (x \right )+7+3 x \right ) \sqrt {21}}{21 x +21}\right )}{21}-\ln \left (x +1\right )-c_{1} = 0 \]

Problem 5792

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

Problem 5793

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

Problem 5794

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

Problem 5795

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

Problem 5796

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

Problem 5797

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

Problem 5798

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

Problem 5799

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

Problem 5800

ODE	\[ \boxed {\frac {2 x y y^{\prime }}{3}-\sqrt {x^{6}-y^{4}}-y^{2}=0} \]
program solution	\[ \frac {\arcsin \left (\frac {y^{2}}{x^{3}}\right )}{2}-\frac {3 \ln \left (x \right )}{2}-c_{1} = 0 \] Verified OK. {0 < x}
Maple solution	\[ -\left (\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{6}-y \left (x \right )^{4}}+y \left (x \right )^{2}}{\sqrt {\textit {\_a}^{6}-y \left (x \right )^{4}}\, \textit {\_a}}d \textit {\_a} \right )+\frac {2 \left (\int _{}^{y \left (x \right )}\frac {\textit {\_f} \left (3 \sqrt {x^{6}-\textit {\_f}^{4}}\, \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{5}}{\left (\textit {\_a}^{6}-\textit {\_f}^{4}\right )^{\frac {3}{2}}}d \textit {\_a} \right )+1\right )}{\sqrt {x^{6}-\textit {\_f}^{4}}}d \textit {\_f} \right )}{3}+c_{1} = 0 \]