2.17.142 Problems 14101 to 14200

Problem 14101

ODE	\[ \boxed {y^{\prime }=\sin \left (x \right )^{4}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 x}{8}+\frac {\sin \left (4 x \right )}{32}-\frac {\sin \left (2 x \right )}{4} \]

Problem 14102

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+\frac {25 y^{\prime \prime }}{2}-5 y^{\prime }+\frac {629 y}{16}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = -1, y^{\prime \prime \prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \left (\frac {37}{208}+\frac {5 i}{104}\right ) {\mathrm e}^{\left (-\frac {1}{2}-3 i\right ) x}+\left (\frac {37}{208}-\frac {5 i}{104}\right ) {\mathrm e}^{\left (-\frac {1}{2}+3 i\right ) x}+\left (-\frac {37}{208}+\frac {111 i}{416}\right ) {\mathrm e}^{\left (\frac {1}{2}-2 i\right ) x}+\left (-\frac {37}{208}-\frac {111 i}{416}\right ) {\mathrm e}^{\left (\frac {1}{2}+2 i\right ) x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (74 \cos \left (3 x \right )+20 \sin \left (3 x \right )\right ) {\mathrm e}^{-\frac {x}{2}}}{208}-\frac {37 \left (\cos \left (2 x \right )-\frac {3 \sin \left (2 x \right )}{2}\right ) {\mathrm e}^{\frac {x}{2}}}{104} \]

Problem 14103

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 y \left (t \right )\\ y^{\prime }\left (t \right )&=-4 x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 0] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 4 \cos \left (4 t \right ) \\ y \left (t \right ) &= -4 \sin \left (4 t \right ) \\ \end{align*}

Problem 14104

ODE	\begin {align*} x^{\prime }\left (t \right )&=-5 x \left (t \right )+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 0] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {4 \,{\mathrm e}^{3 t}}{9}+\frac {32 \,{\mathrm e}^{-6 t}}{9} \\ y \left (t \right ) &= \frac {8 \,{\mathrm e}^{3 t}}{9}-\frac {8 \,{\mathrm e}^{-6 t}}{9} \\ \end{align*}

Problem 14105

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

Problem 14106

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

Problem 14107

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

Problem 14108

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+45 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x} \cos \left (6 x \right )+\frac {c_{2} {\mathrm e}^{3 x} \sin \left (6 x \right )}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (c_{1} \sin \left (6 x \right )+c_{2} \cos \left (6 x \right )\right ) \]

Problem 14109

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x -16 y=0} \]
program solution	\[ y = c_{1} x^{1-\sqrt {17}}+\frac {c_{2} \sqrt {17}\, x^{1+\sqrt {17}}}{34} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (x^{\sqrt {17}} c_{1} +x^{-\sqrt {17}} c_{2} \right ) \]

Problem 14110

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

Problem 14111

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

Problem 14112

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+12 y=2} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{4 x}+\frac {1}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{4 x}+c_{1} {\mathrm e}^{3 x}+\frac {1}{6} \]

Problem 14113

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

Problem 14114

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

Problem 14115

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

Problem 14116

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

Problem 14117

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

Problem 14118

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

Problem 14119

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

Problem 14120

ODE	\[ \boxed {y^{\prime \prime }+4 y=t} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 1, y^{\prime }\left (\frac {\pi }{4}\right ) = \frac {\pi }{16}\right ] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 14121

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

Problem 14122

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

Problem 14123

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

Problem 14124

ODE	\[ \boxed {y^{\prime }-y^{2}=-t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (t \right ) = -\left (\left \{\begin {array}{cc} 0 & t =0 \\ \frac {\left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {t^{2}}{2}\right ) \pi \sqrt {2}-2 \operatorname {BesselK}\left (\frac {3}{4}, \frac {t^{2}}{2}\right )\right ) t}{\pi \sqrt {2}\, \operatorname {BesselI}\left (\frac {1}{4}, \frac {t^{2}}{2}\right )+2 \operatorname {BesselK}\left (\frac {1}{4}, \frac {t^{2}}{2}\right )} & \operatorname {otherwise} \end {array}\right .\right ) \]

Problem 14125

ODE	\[ \boxed {y^{\prime }-\frac {1}{y^{2}}=-t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 14126

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

Problem 14127

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

Problem 14128

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

Problem 14129

ODE	\[ \boxed {y^{\prime }+t y^{2}=4 t^{2}} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
program solution	\[ y = \frac {4 \sqrt {t}\, \operatorname {BesselI}\left (-\frac {3}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}-4 \sqrt {t}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+2 \sqrt {t}\, \operatorname {BesselI}\left (-\frac {3}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )+2 \sqrt {t}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )}{2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselK}\left (\frac {2}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \sqrt {2}+2 \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )-\operatorname {BesselK}\left (\frac {2}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {2 \left (\left (2 \sqrt {2}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right )+\operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (-\frac {3}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right )+\operatorname {BesselK}\left (\frac {3}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \left (-2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right )\right ) \sqrt {t}}{\left (-2 \sqrt {2}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right )-\operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (\frac {2}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right )+\operatorname {BesselK}\left (\frac {2}{5}, \frac {4 t^{\frac {5}{2}}}{5}\right ) \left (-2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right )} \]

Problem 14130

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

Problem 14131

ODE	\[ \boxed {y^{\prime }-6 y^{\frac {2}{3}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = 0 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 14132

ODE	\[ \boxed {y^{\prime }-\sin \left (y\right )=-\cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 0] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 14133

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

Problem 14134

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

Problem 14135

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

Problem 14136

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

Problem 14137

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

Problem 14138

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

Problem 14139

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

Problem 14140

ODE	\[ \boxed {y^{\prime }-\sqrt {25-y^{2}}=0} \] With initial conditions \begin {align*} [y \left (-4\right ) = 3] \end {align*}
program solution	\[ \arcsin \left (\frac {y}{5}\right ) = t +4+\arcsin \left (\frac {3}{5}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 5 \sin \left (t +4+\arcsin \left (\frac {3}{5}\right )\right ) \]

Problem 14141

ODE	\[ \boxed {y^{\prime }-\sqrt {25-y^{2}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5] \end {align*}
program solution	\[ y = 5 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 5 \]

Problem 14142

ODE	\[ \boxed {y^{\prime }-\sqrt {25-y^{2}}=0} \] With initial conditions \begin {align*} [y \left (3\right ) = -6] \end {align*}
program solution	\[ \arcsin \left (\frac {y}{5}\right ) = t -3-\arcsin \left (\frac {6}{5}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 5 \sin \left (t -3-\arcsin \left (\frac {6}{5}\right )\right ) \]

Problem 14143

ODE	\[ \boxed {y^{\prime }-\sqrt {25-y^{2}}=0} \] With initial conditions \begin {align*} [y \left (4\right ) = -5] \end {align*}
program solution	\[ y = -5 \] Verified OK.
Maple solution	\[ y \left (t \right ) = -5 \]

Problem 14144

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

Problem 14145

ODE	\[ \boxed {y^{\prime } t^{3}+t^{4} y=2 t^{3}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = 2 \left (\int _{0}^{t}{\mathrm e}^{\frac {\textit {\_a}^{2}}{2}}d \textit {\_a} \right ) {\mathrm e}^{-\frac {t^{2}}{2}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -i \sqrt {2}\, {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) \]

Problem 14146

ODE	\[ \boxed {2 y^{\prime }+y t=\ln \left (t \right )} \] With initial conditions \begin {align*} [y \left ({\mathrm e}\right ) = 0] \end {align*}
program solution	\[ y = \frac {i {\mathrm e}^{-\frac {t^{2}}{4}} \sqrt {-t^{2}}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i {\mathrm e}}{2}\right )+{\mathrm e}^{-\frac {t^{2}}{4}} \ln \left (t \right ) \sqrt {\pi }\, t \,\operatorname {erf}\left (\frac {\sqrt {-t^{2}}}{2}\right )-{\mathrm e}^{-\frac {t^{2}}{4}} t \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], \frac {t^{2}}{4}\right ) \sqrt {-t^{2}}+{\mathrm e}^{-\frac {t^{2}}{4}} {\mathrm e} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], \frac {{\mathrm e}^{2}}{4}\right ) \sqrt {-t^{2}}}{2 \sqrt {-t^{2}}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\int _{{\mathrm e}}^{t}\ln \left (\textit {\_z1} \right ) {\mathrm e}^{\frac {\textit {\_z1}^{2}}{4}}d \textit {\_z1} \right ) {\mathrm e}^{-\frac {t^{2}}{4}}}{2} \]

Problem 14147

ODE	\[ \boxed {y^{\prime }+y \sec \left (t \right )=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\sin \left (t \right ) \left (\int _{0}^{t}\frac {\textit {\_a} \cos \left (\textit {\_a} \right )}{-1+\sin \left (\textit {\_a} \right )}d \textit {\_a} \right )-\left (\int _{0}^{t}\frac {\textit {\_a} \cos \left (\textit {\_a} \right )}{-1+\sin \left (\textit {\_a} \right )}d \textit {\_a} \right )}{\cos \left (t \right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {i \pi ^{2}+12 i t^{2}+48 i \operatorname {polylog}\left (2, -i {\mathrm e}^{i t}\right )-48 t \ln \left (1+i {\mathrm e}^{i t}\right )-48 \operatorname {Catalan}}{24 \sec \left (t \right )+24 \tan \left (t \right )} \]

Problem 14148

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

Problem 14149

ODE	\[ \boxed {\left (-2+t \right ) y^{\prime }+\left (t^{2}-4\right ) y=\frac {1}{2+t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = \frac {3+\int _{0}^{t}\frac {{\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}}{\textit {\_a}^{2}-4}d \textit {\_a}}{{\mathrm e}^{\frac {t \left (4+t \right )}{2}}-2 \left (\int _{0}^{t}{\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}d \textit {\_a} \right )-\left (\int _{0}^{t}\textit {\_a} \,{\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}d \textit {\_a} \right )+\int _{0}^{t}\frac {\textit {\_a}^{3} {\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}}{\textit {\_a}^{2}-4}d \textit {\_a} +2 \left (\int _{0}^{t}\frac {\textit {\_a}^{2} {\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}}{\textit {\_a}^{2}-4}d \textit {\_a} \right )-4 \left (\int _{0}^{t}\frac {\textit {\_a} \,{\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}}{\textit {\_a}^{2}-4}d \textit {\_a} \right )-8 \left (\int _{0}^{t}\frac {{\mathrm e}^{\frac {\textit {\_a} \left (4+\textit {\_a} \right )}{2}}}{\textit {\_a}^{2}-4}d \textit {\_a} \right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\int _{0}^{t}\frac {{\mathrm e}^{\frac {\textit {\_z1} \left (\textit {\_z1} +4\right )}{2}}}{\textit {\_z1}^{2}-4}d \textit {\_z1} +3\right ) {\mathrm e}^{-\frac {t \left (t +4\right )}{2}} \]

Problem 14150

ODE	\[ \boxed {y^{\prime }+\frac {y}{\sqrt {-t^{2}+4}}=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ -\left (\int _{0}^{t}-\frac {\left (-\textit {\_a} \sqrt {-\textit {\_a}^{2}+4}+y\right ) {\mathrm e}^{\arcsin \left (\frac {\textit {\_a}}{2}\right )}}{\sqrt {-\textit {\_a}^{2}+4}}d \textit {\_a} \right )-y \left (\int _{0}^{t}\frac {{\mathrm e}^{\arcsin \left (\frac {\textit {\_a}}{2}\right )}}{\sqrt {-\textit {\_a}^{2}+4}}d \textit {\_a} \right )+{\mathrm e}^{\arcsin \left (\frac {t}{2}\right )} y = 0 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\int _{0}^{t}\textit {\_z1} \,{\mathrm e}^{\arcsin \left (\frac {\textit {\_z1}}{2}\right )}d \textit {\_z1} \right ) {\mathrm e}^{-\arcsin \left (\frac {t}{2}\right )} \]

Problem 14151

ODE	\[ \boxed {y^{\prime }+\frac {y}{\sqrt {-t^{2}+4}}=t} \] With initial conditions \begin {align*} [y \left (3\right ) = -1] \end {align*}
program solution	\[ -\left (\int _{3}^{t}-\frac {\left (-\textit {\_a} \sqrt {-\textit {\_a}^{2}+4}+y\right ) {\mathrm e}^{\arcsin \left (\frac {\textit {\_a}}{2}\right )}}{\sqrt {-\textit {\_a}^{2}+4}}d \textit {\_a} \right )-y \left (\int _{3}^{t}\frac {{\mathrm e}^{\arcsin \left (\frac {\textit {\_a}}{2}\right )}}{\sqrt {-\textit {\_a}^{2}+4}}d \textit {\_a} \right )+{\mathrm e}^{\arcsin \left (\frac {t}{2}\right )} y = -{\mathrm e}^{\arcsin \left (\frac {3}{2}\right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\int _{3}^{t}\textit {\_z1} \,{\mathrm e}^{\arcsin \left (\frac {\textit {\_z1}}{2}\right )}d \textit {\_z1} -{\mathrm e}^{\arcsin \left (\frac {3}{2}\right )}\right ) {\mathrm e}^{-\arcsin \left (\frac {t}{2}\right )} \]

Problem 14152

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

Problem 14153

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

Problem 14154

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

Problem 14155

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

Problem 14156

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

Problem 14157

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

Problem 14158

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

Problem 14159

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

Problem 14160

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

Problem 14161

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

Problem 14162

ODE	\[ \boxed {\left (5+\frac {9}{y^{8}}\right ) y^{\prime }=-4 t^{3}-6} \]
program solution	\[ -\frac {t^{4}}{2}-3 t -\frac {5 y}{2}+\frac {9}{14 y^{7}} = c_{1} \] Verified OK.
Maple solution	\[ \frac {t^{4}}{2}+3 t +\frac {5 y \left (t \right )}{2}-\frac {9}{14 y \left (t \right )^{7}}+c_{1} = 0 \]

Problem 14163

ODE	\[ \boxed {\left (9+\frac {1}{s^{2}}-4 s^{8}\right ) s^{\prime }=-\frac {6}{t^{9}}+\frac {6}{t^{3}}-t^{7}} \]
program solution	\[ \frac {-t^{16}-24 t^{6}+6}{8 t^{8}}+\frac {4 s^{9}}{9}-9 s+\frac {1}{s} = c_{1} \] Verified OK.
Maple solution	\[ \frac {t^{8}}{8}+\frac {3}{t^{2}}-\frac {3}{4 t^{8}}-\frac {4 s \left (t \right )^{9}}{9}+9 s \left (t \right )-\frac {1}{s \left (t \right )}+c_{1} = 0 \]

Problem 14164

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

Problem 14165

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

Problem 14166

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

Problem 14167

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

Problem 14168

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

Problem 14169

ODE	\[ \boxed {\cos \left (y\right ) y^{\prime }=8 \sin \left (8 t \right )} \]
program solution	\[ \frac {\cos \left (8 t \right )}{8}+\frac {\sin \left (y\right )}{8} = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \arcsin \left (-\cos \left (8 t \right )+64 c_{1} \right ) \]

Problem 14170

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

Problem 14171

ODE	\[ \boxed {\left (5 x^{5}-4 \cos \left (x\right )\right ) x^{\prime }=-2 \cos \left (9 t \right )-2 \sin \left (7 t \right )} \]
program solution	\[ -\frac {\sin \left (9 t \right )}{9}+\frac {\cos \left (7 t \right )}{7}-\frac {5 x^{6}}{12}+2 \sin \left (x\right ) = c_{1} \] Verified OK.
Maple solution	\[ \frac {\sin \left (9 t \right )}{9}-\frac {\cos \left (7 t \right )}{7}+\frac {5 x \left (t \right )^{6}}{12}-2 \sin \left (x \left (t \right )\right )+c_{1} = 0 \]

Problem 14172

ODE	\[ \boxed {20 \sinh \left (y\right ) y^{\prime }=-\cosh \left (6 t \right )-5 \sinh \left (4 t \right )} \]
program solution	\[ y = \operatorname {arccosh}\left (-\frac {\sinh \left (6 t \right )}{120}-\frac {\cosh \left (4 t \right )}{16}-\frac {c_{1}}{20}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {arccosh}\left (-\frac {\cosh \left (4 t \right )}{16}-\frac {\sinh \left (6 t \right )}{120}-\frac {c_{1}}{20}\right ) \]

Problem 14173

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

Problem 14174

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

Problem 14175

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

Problem 14176

ODE	\[ \boxed {-\left (\cos \left (4 y\right )-4 \cos \left (y\right )\right ) y^{\prime }=-3 \sin \left (t \right )+\sin \left (3 t \right )} \]
program solution	\[ \frac {4 \cos \left (t \right )^{3}}{3}-4 \cos \left (t \right )+4 \sin \left (y\right )-\frac {\sin \left (4 y\right )}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \arctan \left (\frac {4 \cos \left (t \right )^{3}-12 c_{1} -12 \cos \left (t \right )}{6 \operatorname {RootOf}\left (36 \textit {\_Z}^{8}-72 \textit {\_Z}^{6}+16 \cos \left (t \right )^{6}-144 \textit {\_Z}^{5}-96 c_{1} \cos \left (t \right )^{3}+45 \textit {\_Z}^{4}-96 \cos \left (t \right )^{4}+216 \textit {\_Z}^{3}+144 c_{1}^{2}+288 \cos \left (t \right ) c_{1} +135 \textit {\_Z}^{2}+144 \cos \left (t \right )^{2}-72 \textit {\_Z} -144\right )^{3}-3 \operatorname {RootOf}\left (36 \textit {\_Z}^{8}-72 \textit {\_Z}^{6}+16 \cos \left (t \right )^{6}-144 \textit {\_Z}^{5}-96 c_{1} \cos \left (t \right )^{3}+45 \textit {\_Z}^{4}-96 \cos \left (t \right )^{4}+216 \textit {\_Z}^{3}+144 c_{1}^{2}+288 \cos \left (t \right ) c_{1} +135 \textit {\_Z}^{2}+144 \cos \left (t \right )^{2}-72 \textit {\_Z} -144\right )-12}, \operatorname {RootOf}\left (36 \textit {\_Z}^{8}-72 \textit {\_Z}^{6}+16 \cos \left (t \right )^{6}-144 \textit {\_Z}^{5}-96 c_{1} \cos \left (t \right )^{3}+45 \textit {\_Z}^{4}-96 \cos \left (t \right )^{4}+216 \textit {\_Z}^{3}+144 c_{1}^{2}+288 \cos \left (t \right ) c_{1} +135 \textit {\_Z}^{2}+144 \cos \left (t \right )^{2}-72 \textit {\_Z} -144\right )\right ) \]

Problem 14177

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

Problem 14178

ODE	\[ \boxed {\left (2-\frac {5}{y^{2}}\right ) y^{\prime }=-4 \cos \left (x \right )^{2}} \]
program solution	\[ -\frac {\sin \left (2 x \right )}{4}-\frac {x}{2}-\frac {y}{2}-\frac {5}{4 y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -c_{1} -\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}-\frac {\sqrt {-158+16 \left (x +2 c_{1} \right ) \sin \left (2 x \right )+16 x^{2}+64 c_{1} x +64 c_{1}^{2}-2 \cos \left (4 x \right )}}{8} \\ y \left (x \right ) &= -c_{1} -\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}+\frac {\sqrt {-158+16 \left (x +2 c_{1} \right ) \sin \left (2 x \right )+16 x^{2}+64 c_{1} x +64 c_{1}^{2}-2 \cos \left (4 x \right )}}{8} \\ \end{align*}

Problem 14179

ODE	\[ \boxed {y^{\prime }-\frac {t^{3}}{y \sqrt {\left (1-y^{2}\right ) \left (t^{4}+9\right )}}=0} \]
program solution	\[ -\frac {\left (t^{4}+9\right ) \sqrt {y-1}\, \sqrt {y+1}}{2 \sqrt {-\left (y^{2}-1\right ) \left (t^{4}+9\right )}}+\frac {\left (y-1\right )^{\frac {3}{2}} \left (y+1\right )^{\frac {3}{2}}}{3} = c_{1} \] Verified OK.
Maple solution	\[ -\left (-\frac {y \left (t \right )^{2}}{3}+\int _{}^{t}\frac {\textit {\_a}^{3}}{\sqrt {-\left (\textit {\_a}^{4}+9\right ) \left (y \left (t \right )^{2}-1\right )}}d \textit {\_a} +\frac {1}{3}\right ) \sqrt {y \left (t \right )+1}\, \sqrt {-1+y \left (t \right )}+c_{1} = 0 \]

Problem 14180

ODE	\[ \boxed {\tan \left (y\right ) \sec \left (y\right )^{2} y^{\prime }=-\cos \left (2 x \right )^{3} \sin \left (2 x \right )} \]
program solution	\[ \frac {\cos \left (2 x \right )^{4}}{8}-\frac {\sec \left (y\right )^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \operatorname {arccot}\left (\frac {8}{\sqrt {-2+4 \cos \left (4 x \right )^{2}-256 c_{1} +8 \cos \left (4 x \right )}}\right ) \\ y \left (x \right ) &= \frac {\pi }{2}+\arctan \left (\frac {8}{\sqrt {-2+4 \cos \left (4 x \right )^{2}-256 c_{1} +8 \cos \left (4 x \right )}}\right ) \\ \end{align*}

Problem 14181

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

Problem 14182

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

Problem 14183

ODE	\[ \boxed {-\frac {\left (1-\frac {1}{y}\right )^{2} y^{\prime }}{y^{2}}=-\frac {x -2}{x^{2}-4 x +3}} \]
program solution	\[ -\frac {\ln \left (\left (x -1\right ) \left (x -3\right )\right )}{2}-\frac {1}{y}+\frac {1}{y^{2}}-\frac {1}{3 y^{3}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (\left (9 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )^{2}+36 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) c_{1} +36 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) c_{1}^{2}-9 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )^{2}+6 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-36 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) c_{1} +12 \,\operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right ) c_{1} -36 c_{1}^{2}-18 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-36 c_{1} -8\right )^{\frac {2}{3}}+6 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-2 \,36^{\frac {1}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}}+12 c_{1} +4\right ) 36^{\frac {2}{3}}}{36 \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}} \left (6 c_{1} +3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )\right )} \\ y \left (x \right ) &= \frac {\left (-i 36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}} \sqrt {3}+6 i \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) \sqrt {3}+12 i \sqrt {3}\, c_{1} +4 i \sqrt {3}-36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}}-6 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-4 \,36^{\frac {1}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}}-12 c_{1} -4\right ) 36^{\frac {2}{3}}}{216 \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}} \left (\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+2 c_{1} \right )} \\ y \left (x \right ) &= \frac {\left (i 36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}} \sqrt {3}-6 i \ln \left (\left (-1+x \right ) \left (-3+x \right )\right ) \sqrt {3}-12 i \sqrt {3}\, c_{1} -4 i \sqrt {3}-36^{\frac {2}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {2}{3}}-6 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )-4 \,36^{\frac {1}{3}} \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}}-12 c_{1} -4\right ) 36^{\frac {2}{3}}}{216 \left (\left (\left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} \right ) \operatorname {csgn}\left (3 \ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+6 c_{1} +2\right )-c_{1} -\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}-\frac {2}{3}\right ) \left (\frac {\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )}{2}+c_{1} +\frac {1}{3}\right )\right )^{\frac {1}{3}} \left (\ln \left (\left (-1+x \right ) \left (-3+x \right )\right )+2 c_{1} \right )} \\ \end{align*}

Problem 14184

ODE	\[ \boxed {\frac {\cos \left (y\right ) y^{\prime }}{\left (1-\sin \left (y\right )\right )^{2}}=\sin \left (x \right )^{3} \cos \left (x \right )} \]
program solution	\[ -\frac {\sin \left (x \right )^{4}}{4}-\frac {1}{\sin \left (y\right )-1} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arcsin \left (\frac {\cos \left (2 x \right )^{2}+16 c_{1} -2 \cos \left (2 x \right )-15}{\cos \left (2 x \right )^{2}+16 c_{1} -2 \cos \left (2 x \right )+1}\right ) \]

Problem 14185

ODE	\[ \boxed {y^{\prime }-\frac {\left (5-2 \cos \left (x \right )\right )^{3} \sin \left (x \right ) \cos \left (y\right )^{4}}{\sin \left (y\right )}=0} \]
program solution	\[ \frac {\left (2 \cos \left (x \right )-5\right )^{4}}{8}-\frac {\sec \left (y\right )^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \pi -\arccos \left (\frac {2 \,3^{\frac {2}{3}} \left (\left (8 c_{1} -931-2 \cos \left (4 x \right )-308 \cos \left (2 x \right )+40 \cos \left (3 x \right )+1120 \cos \left (x \right )\right )^{2}\right )^{\frac {1}{3}}}{-48 \cos \left (x \right )^{4}+480 \cos \left (x \right )^{3}-1800 \cos \left (x \right )^{2}+24 c_{1} +3000 \cos \left (x \right )-1875}\right ) \\ y \left (x \right ) &= \frac {\pi }{2}-i \operatorname {arcsinh}\left (\frac {\left (\frac {i 3^{\frac {2}{3}}}{3}+3^{\frac {1}{6}}\right ) \left (\left (8 c_{1} -931-2 \cos \left (4 x \right )-308 \cos \left (2 x \right )+40 \cos \left (3 x \right )+1120 \cos \left (x \right )\right )^{2}\right )^{\frac {1}{3}}}{16 \cos \left (x \right )^{4}-160 \cos \left (x \right )^{3}+600 \cos \left (x \right )^{2}-1000 \cos \left (x \right )-8 c_{1} +625}\right ) \\ y \left (x \right ) &= \frac {\pi }{2}+i \operatorname {arcsinh}\left (\frac {\left (\left (8 c_{1} -931-2 \cos \left (4 x \right )-308 \cos \left (2 x \right )+40 \cos \left (3 x \right )+1120 \cos \left (x \right )\right )^{2}\right )^{\frac {1}{3}} 3^{\frac {2}{3}} \left (i-\sqrt {3}\right )}{-48 \cos \left (x \right )^{4}+480 \cos \left (x \right )^{3}-1800 \cos \left (x \right )^{2}+24 c_{1} +3000 \cos \left (x \right )-1875}\right ) \\ \end{align*}

Problem 14186

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

Problem 14187

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

Problem 14188

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

Problem 14189

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

Problem 14190

ODE	\[ \boxed {4 \left (x -1\right )^{2} y^{\prime }-3 \left (y+3\right )^{2}=0} \]
program solution	\[ y = \frac {\left (4 x -13\right ) c_{3} -5 x -4}{3 x +3 c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-9 x +9\right ) c_{1} -4 x +13}{-3+\left (3 x -3\right ) c_{1}} \]

Problem 14191

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

Problem 14192

ODE	\[ \boxed {y^{\prime }-y^{3}=1} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3}+1}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{2}+\frac {\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )+6 \sqrt {3}\, c_{1} +6 \sqrt {3}\, t -6 \textit {\_Z} \right )\right )}{2} \]

Problem 14193

ODE	\[ \boxed {y^{\prime }-y^{3}=-1} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3}-1}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {1}{2}+\frac {\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (\tan \left (\textit {\_Z} \right )-\sqrt {3}\right )+6 \sqrt {3}\, c_{1} +6 \sqrt {3}\, t +6 \textit {\_Z} \right )\right )}{2} \]

Problem 14194

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

Problem 14195

ODE	\[ \boxed {y^{\prime }-y^{3}+y^{2}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3}-\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{t -1}\right )+1} \]

Problem 14196

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

Problem 14197

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

Problem 14198

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

Problem 14199

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

Problem 14200

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