Problem 1
| ODE |
\[ \boxed {y^{\prime }=1+2 x} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
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program solution |
\[ y = x^{2}+x +3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{2}+x +3 \] |
Problem 2
| ODE |
\[ \boxed {y^{\prime }=\left (-2+x \right )^{2}} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {1}{3} x^{3}-2 x^{2}+4 x -\frac {5}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-2+x \right )^{3}}{3}+1 \] |
Problem 3
| ODE |
\[ \boxed {y^{\prime }=\sqrt {x}} \] With initial conditions \begin {align*} [y \left (4\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {2 x^{\frac {3}{2}}}{3}-\frac {16}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 x^{\frac {3}{2}}}{3}-\frac {16}{3} \] |
Problem 4
| ODE |
\[ \boxed {y^{\prime }=\frac {1}{x^{2}}} \] With initial conditions \begin {align*} [y \left (1\right ) = 5] \end {align*}
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program solution |
\[ y = \frac {-1+6 x}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {1}{x}+6 \] |
Problem 5
| ODE |
\[ \boxed {y^{\prime }=\frac {1}{\sqrt {2+x}}} \] With initial conditions \begin {align*} [y \left (2\right ) = -1] \end {align*}
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program solution |
\[ y = 2 \sqrt {2+x}-5 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 \sqrt {2+x}-5 \] |
Problem 6
| ODE |
\[ \boxed {y^{\prime }=x \sqrt {x^{2}+9}} \] With initial conditions \begin {align*} [y \left (-4\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {\left (x^{2}+9\right )^{\frac {3}{2}}}{3}-\frac {125}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{2} \sqrt {x^{2}+9}}{3}+3 \sqrt {x^{2}+9}-\frac {125}{3} \] |
Problem 7
| ODE |
\[ \boxed {y^{\prime }=\frac {10}{x^{2}+1}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = 10 \arctan \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 10 \arctan \left (x \right ) \] |
Problem 8
| ODE |
\[ \boxed {y^{\prime }=\cos \left (2 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {\sin \left (2 x \right )}{2}+1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sin \left (2 x \right )}{2}+1 \] |
Problem 9
| ODE |
\[ \boxed {y^{\prime }=\frac {1}{\sqrt {-x^{2}+1}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \arcsin \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arcsin \left (x \right ) \] |
Problem 10
| ODE |
\[ \boxed {y^{\prime }=x \,{\mathrm e}^{-x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = -x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}+2 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2+\left (-x -1\right ) {\mathrm e}^{-x} \] |
Problem 11
| ODE |
\[ \boxed {y^{\prime }+y=-\sin \left (x \right )} \] |
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program solution |
\[ y = -\frac {{\mathrm e}^{-x} \left (\sin \left (x \right ) {\mathrm e}^{x}-\cos \left (x \right ) {\mathrm e}^{x}-2 c_{1} \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2}+{\mathrm e}^{-x} c_{1} \] |
Problem 12
| ODE |
\[ \boxed {y^{\prime }-y=x} \] |
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program solution |
\[ y = -\left (x \,{\mathrm e}^{-x}+{\mathrm e}^{-x}-c_{1} \right ) {\mathrm e}^{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -x -1+{\mathrm e}^{x} c_{1} \] |
Problem 13
| ODE |
\[ \boxed {y^{\prime }-y=-\sin \left (x \right )} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{x} \left (\sin \left (x \right ) {\mathrm e}^{-x}+\cos \left (x \right ) {\mathrm e}^{-x}+2 c_{1} \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}+{\mathrm e}^{x} c_{1} \] |
Problem 14
| ODE |
\[ \boxed {y^{\prime }+y=x} \] |
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program solution |
\[ y = \left (x \,{\mathrm e}^{x}-{\mathrm e}^{x}+c_{1} \right ) {\mathrm e}^{-x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x -1+{\mathrm e}^{-x} c_{1} \] |
Problem 15
| ODE |
\[ \boxed {y^{\prime }-y=1-x} \] |
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program solution |
\[ y = \left (x \,{\mathrm e}^{-x}+c_{1} \right ) {\mathrm e}^{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x +{\mathrm e}^{x} c_{1} \] |
Problem 16
| ODE |
\[ \boxed {y^{\prime }+y=x +1} \] |
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program solution |
\[ y = \left (x \,{\mathrm e}^{x}+c_{1} \right ) {\mathrm e}^{-x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x +{\mathrm e}^{-x} c_{1} \] |
Problem 17
| ODE |
\[ \boxed {y^{\prime }+y=x^{2}} \] |
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program solution |
\[ y = \left (x^{2} {\mathrm e}^{x}-2 x \,{\mathrm e}^{x}+2 \,{\mathrm e}^{x}+c_{1} \right ) {\mathrm e}^{-x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{2}-2 x +2+{\mathrm e}^{-x} c_{1} \] |
Problem 18
| ODE |
\[ \boxed {y^{\prime }+y=x^{2}-2} \] |
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program solution |
\[ y = \left (x^{2} {\mathrm e}^{x}-2 x \,{\mathrm e}^{x}+c_{1} \right ) {\mathrm e}^{-x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{2}-2 x +{\mathrm e}^{-x} c_{1} \] |
Problem 19
| ODE |
\[ \boxed {y^{\prime }-2 x^{2} y^{2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
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program solution |
\[ y = -\frac {3}{2 x^{3}+1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {3}{2 x^{3}+1} \] |
Problem 20
| ODE |
\[ \boxed {y^{\prime }-x \ln \left (y\right )=0} \] |
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program solution |
\[ -\frac {x^{2}}{2}-\operatorname {expIntegral}_{1}\left (-\ln \left (y\right )\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (x^{2}+2 \,\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right )+2 c_{1} \right )} \] |
Problem 21
| ODE |
\[ \boxed {y^{\prime }-y^{\frac {1}{3}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ \frac {3 y^{\frac {2}{3}}}{2} = x +\frac {3}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (2 x +3\right ) \sqrt {6 x +9}}{9} \] |
Problem 22
| ODE |
\[ \boxed {y^{\prime }-y^{\frac {1}{3}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = 0 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 0 \] |
Problem 23
| ODE |
\[ \boxed {y y^{\prime }=x -1} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ -\frac {x^{2}}{2}+\frac {y^{2}}{2}+x = {\frac {1}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 1-x \] |
Problem 24
| ODE |
\[ \boxed {y y^{\prime }=x -1} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
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program solution |
\[ -\frac {x^{2}}{2}+\frac {y^{2}}{2}+x = {\frac {1}{2}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 1-x \\ y \left (x \right ) &= x -1 \\ \end{align*} |
Problem 25
| ODE |
\[ \boxed {y^{\prime }-\ln \left (1+y^{2}\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = 0 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 0 \] |
Problem 26
| ODE |
\[ \boxed {y^{\prime }+y^{2}=x^{2}} \] |
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program solution |
\[ y = \frac {x \left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_{3} -\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x \left (\operatorname {BesselI}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_{1} -\operatorname {BesselK}\left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{c_{1} \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \] |
Problem 27
| ODE |
\[ \boxed {2 y x +y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{-x^{2}-2 c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}} \] |
Problem 28
| ODE |
\[ \boxed {2 x y^{2}+y^{\prime }=0} \] |
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program solution |
\[ y = \frac {1}{x^{2}+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{x^{2}+c_{1}} \] |
Problem 29
| ODE |
\[ \boxed {y^{\prime }-\sin \left (x \right ) y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\cos \left (x \right )+c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\cos \left (x \right )} \] |
Problem 30
| ODE |
\[ \boxed {\left (x +1\right ) y^{\prime }-4 y=0} \] |
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program solution |
\[ y = {\mathrm e}^{4 c_{1}} \left (x +1\right )^{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (x +1\right )^{4} \] |
Problem 31
| ODE |
\[ \boxed {2 \sqrt {x}\, y^{\prime }-\sqrt {1-y^{2}}=0} \] |
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program solution |
\[ y = \sin \left (\sqrt {x}+\frac {c_{1}}{2}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sin \left (\sqrt {x}+\frac {c_{1}}{2}\right ) \] |
Problem 32
| ODE |
\[ \boxed {y^{\prime }-3 \sqrt {y x}=0} \] |
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program solution |
\[ -\frac {2 x \sqrt {y x}}{\sqrt {y}}+2 \sqrt {y} = c_{1} \] Verified OK.
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Maple solution | \[ \frac {\left (c_{1} x^{3}-y \left (x \right ) c_{1} +1\right ) \sqrt {x y \left (x \right )}-x^{2} \left (c_{1} x^{3}-y \left (x \right ) c_{1} -1\right )}{\left (x^{3}-y \left (x \right )\right ) \left (x^{2}-\sqrt {x y \left (x \right )}\right )} = 0 \] |
Problem 33
| ODE |
\[ \boxed {y^{\prime }-4 \left (y x \right )^{\frac {1}{3}}=0} \] |
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program solution |
\[ -\frac {3 x \left (y x \right )^{\frac {1}{3}}}{y^{\frac {1}{3}}}+\frac {3 y^{\frac {2}{3}}}{2} = c_{1} \] Verified OK.
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Maple solution | \[ -\frac {32 x \left (\left (-c_{1} x^{5}+\frac {y \left (x \right )^{2} c_{1} x}{8}+\frac {x}{16}\right ) \left (x y \left (x \right )\right )^{\frac {2}{3}}+\left (c_{1} x^{4}-\frac {y \left (x \right )^{2} c_{1}}{8}+\frac {1}{8}\right ) \left (x^{3}+\frac {y \left (x \right ) \left (x y \left (x \right )\right )^{\frac {1}{3}}}{4}\right )\right )}{\left (8 x^{4}-y \left (x \right )^{2}\right ) \left (-\left (x y \left (x \right )\right )^{\frac {2}{3}}+2 x^{2}\right )^{2}} = 0 \] |
Problem 34
| ODE |
\[ \boxed {y^{\prime }-2 x \sec \left (y\right )=0} \] |
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program solution |
\[ -\frac {x^{2}}{2}+\frac {\sin \left (y\right )}{2} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arcsin \left (x^{2}+2 c_{1} \right ) \] |
Problem 35
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-2 y=0} \] |
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program solution |
\[ y = {\mathrm e}^{2 \,\operatorname {arctanh}\left (x \right )-2 c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (x +1\right ) c_{1}}{x -1} \] |
Problem 36
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }-\left (1+y\right )^{2}=0} \] |
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program solution |
\[ y = \frac {-\arctan \left (x \right )-c_{3} -1}{\arctan \left (x \right )+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\arctan \left (x \right )-c_{1} -1}{\arctan \left (x \right )+c_{1}} \] |
Problem 37
| ODE |
\[ \boxed {y^{\prime }-x y^{3}=0} \] |
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program solution |
\[ -\frac {x^{2}}{2}-\frac {1}{2 y^{2}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {-x^{2}+c_{1}}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {-x^{2}+c_{1}}} \\ \end{align*} |
Problem 38
| ODE |
\[ \boxed {y y^{\prime }-x \left (1+y^{2}\right )=0} \] |
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program solution |
\[ -\frac {x^{2}}{2}+\frac {\ln \left (1+y^{2}\right )}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {{\mathrm e}^{x^{2}} c_{1} -1} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{x^{2}} c_{1} -1} \\ \end{align*} |
Problem 39
| ODE |
\[ \boxed {y^{\prime }-\frac {1+\sqrt {x}}{1+\sqrt {y}}=0} \] |
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program solution |
\[ -\frac {2 x^{\frac {3}{2}}}{3}-x +\frac {2 y^{\frac {3}{2}}}{3}+y = c_{1} \] Verified OK.
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Maple solution | \[ x +\frac {2 x^{\frac {3}{2}}}{3}-y \left (x \right )-\frac {2 y \left (x \right )^{\frac {3}{2}}}{3}+c_{1} = 0 \] |
Problem 40
| ODE |
\[ \boxed {y^{\prime }-\frac {\left (x -1\right ) y^{5}}{x^{2} \left (-y+2 y^{3}\right )}=0} \] |
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program solution |
\[ -\frac {1}{x}-\ln \left (x \right )-\frac {2}{y}+\frac {1}{3 y^{3}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {8 x^{2} 2^{\frac {1}{3}}-4 x \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {2}{3}}}{\left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}} \left (6 c_{1} x +6 x \ln \left (x \right )+6\right )} \\ y \left (x \right ) &= -\frac {8 x \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}}-8 x^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}+2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {2}{3}}}{\left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}} \left (12 c_{1} x +12 x \ln \left (x \right )+12\right )} \\ y \left (x \right ) &= \frac {-8 x \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}}-8 x^{2} \left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}}+2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {2}{3}}}{\left (3 x \left (x \ln \left (x \right )+c_{1} x +1\right ) \sqrt {9+9 \ln \left (x \right )^{2} x^{2}+18 \left (c_{1} x^{2}+x \right ) \ln \left (x \right )+\left (9 c_{1}^{2}-32\right ) x^{2}+18 c_{1} x}+9 \left (x \ln \left (x \right )+1+\left (c_{1} +\frac {4}{3}\right ) x \right ) x \left (x \ln \left (x \right )+1+\left (c_{1} -\frac {4}{3}\right ) x \right )\right )^{\frac {1}{3}} \left (12 c_{1} x +12 x \ln \left (x \right )+12\right )} \\ \end{align*} |
Problem 41
| ODE |
\[ \boxed {\left (x^{2}+1\right ) \tan \left (y\right ) y^{\prime }=x} \] |
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program solution |
\[ -\frac {\ln \left (x^{2}+1\right )}{2}-\ln \left (\cos \left (y\right )\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arccos \left (\frac {1}{\sqrt {x^{2}+1}\, c_{1}}\right ) \] |
Problem 42
| ODE |
\[ \boxed {y^{\prime }-y-y x=x +1} \] |
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program solution |
\[ y = {\mathrm e}^{\frac {1}{2} x^{2}+x +c_{1}}-1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -1+{\mathrm e}^{\frac {x \left (2+x \right )}{2}} c_{1} \] |
Problem 43
| ODE |
\[ \boxed {y^{\prime } x^{2}-y^{2}+x^{2} y^{2}=-x^{2}+1} \] |
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program solution |
\[ y = \frac {-\sin \left (\frac {x^{2}+1}{x}\right )+c_{3} \cos \left (\frac {x^{2}+1}{x}\right )}{c_{3} \sin \left (\frac {x^{2}+1}{x}\right )+\cos \left (\frac {x^{2}+1}{x}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\tan \left (\frac {c_{1} x +x^{2}+1}{x}\right ) \] |
Problem 44
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{x} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2 \,{\mathrm e}] \end {align*}
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program solution |
\[ y = 2 \,{\mathrm e}^{{\mathrm e}^{x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 \,{\mathrm e}^{{\mathrm e}^{x}} \] |
Problem 45
| ODE |
\[ \boxed {y^{\prime }-3 \left (1+y^{2}\right ) x^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {-\sin \left (x^{3}\right )-\cos \left (x^{3}\right )}{\sin \left (x^{3}\right )-\cos \left (x^{3}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tan \left (x^{3}+\frac {\pi }{4}\right ) \] |
Problem 46
| ODE |
\[ \boxed {2 y y^{\prime }=\frac {x}{\sqrt {x^{2}-16}}} \] With initial conditions \begin {align*} [y \left (5\right ) = 2] \end {align*}
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program solution |
\[ y^{2}-\sqrt {x^{2}-16} = 1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sqrt {\sqrt {x^{2}-16}\, \left (x^{2}+\sqrt {x^{2}-16}-16\right )}}{\sqrt {x^{2}-16}} \] |
Problem 47
| ODE |
\[ \boxed {y^{\prime }+y-4 y x^{3}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -3] \end {align*}
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program solution |
\[ y = -3 \,{\mathrm e}^{x \left (x -1\right ) \left (x^{2}+x +1\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -3 \,{\mathrm e}^{x \left (x -1\right ) \left (x^{2}+x +1\right )} \] |
Problem 48
| ODE |
\[ \boxed {y^{\prime }-2 y=-1} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ -\frac {\ln \left (2\right )}{2}+\frac {\ln \left (-1+2 y\right )}{2} = x -1-\frac {\ln \left (2\right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{2}+\frac {{\mathrm e}^{2 x -2}}{2} \] |
Problem 49
| ODE |
\[ \boxed {\tan \left (x \right ) y^{\prime }-y=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = \frac {\pi }{2}\right ] \end {align*}
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program solution |
\[ y = \frac {\pi \sin \left (x \right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\pi \sin \left (x \right )}{2} \] |
Problem 50
| ODE |
\[ \boxed {-y+y^{\prime } x -2 x^{2} y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ y = {\mathrm e}^{x^{2}-1} x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \,{\mathrm e}^{\left (x -1\right ) \left (x +1\right )} \] |
Problem 51
| ODE |
\[ \boxed {y^{\prime }-2 x y^{2}-3 x^{2} y^{2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
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program solution |
\[ y = -\frac {1}{x^{3}+x^{2}-1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {1}{x^{3}+x^{2}-1} \] |
Problem 52
| ODE |
\[ \boxed {y^{\prime }-6 \,{\mathrm e}^{2 x -y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \ln \left (3 \,{\mathrm e}^{2 x}-2\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \ln \left (-2+3 \,{\mathrm e}^{2 x}\right ) \] |
Problem 53
| ODE |
\[ \boxed {2 \sqrt {x}\, y^{\prime }-\cos \left (y\right )^{2}=0} \] With initial conditions \begin {align*} \left [y \left (4\right ) = \frac {\pi }{4}\right ] \end {align*}
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program solution |
\[ -2 \sqrt {x}+2 \tan \left (y\right ) = -2 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arctan \left (-1+\sqrt {x}\right ) \] |
Problem 54
| ODE |
\[ \boxed {y^{\prime }+y=2} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ -\ln \left (y-2\right ) = x -\ln \left (2\right )-i \pi \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2-2 \,{\mathrm e}^{-x} \] |
Problem 55
| ODE |
\[ \boxed {y^{\prime }-2 y=3 \,{\mathrm e}^{2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = 3 x \,{\mathrm e}^{2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 3 \,{\mathrm e}^{2 x} x \] |
Problem 56
| ODE |
\[ \boxed {3 y+y^{\prime }=2 x \,{\mathrm e}^{-3 x}} \] |
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program solution |
\[ y = {\mathrm e}^{-3 x} \left (x^{2}+c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (x^{2}+c_{1} \right ) {\mathrm e}^{-3 x} \] |
Problem 57
| ODE |
\[ \boxed {-2 y x +y^{\prime }={\mathrm e}^{x^{2}}} \] |
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program solution |
\[ y = {\mathrm e}^{x^{2}} \left (x +c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} +x \right ) {\mathrm e}^{x^{2}} \] |
Problem 58
| ODE |
\[ \boxed {2 y+y^{\prime } x=3 x} \] With initial conditions \begin {align*} [y \left (1\right ) = 5] \end {align*}
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program solution |
\[ y = \frac {x^{3}+4}{x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x +\frac {4}{x^{2}} \] |
Problem 59
| ODE |
\[ \boxed {y+2 y^{\prime } x=10 \sqrt {x}} \] With initial conditions \begin {align*} [y \left (2\right ) = 5] \end {align*}
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program solution |
\[ y = \frac {5 x -10+5 \sqrt {2}}{\sqrt {x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-10+5 \sqrt {2}+5 x}{\sqrt {x}} \] |
Problem 60
| ODE |
\[ \boxed {y+2 y^{\prime } x=10 \sqrt {x}} \] |
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program solution |
\[ y = \frac {10 x +c_{1}}{2 \sqrt {x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {5 x +c_{1}}{\sqrt {x}} \] |
Problem 61
| ODE |
\[ \boxed {y+3 y^{\prime } x=12 x} \] |
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program solution |
\[ y = \frac {9 x^{\frac {4}{3}}+c_{1}}{3 x^{\frac {1}{3}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 3 x +\frac {c_{1}}{x^{\frac {1}{3}}} \] |
Problem 62
| ODE |
\[ \boxed {-y+y^{\prime } x=x} \] With initial conditions \begin {align*} [y \left (1\right ) = 7] \end {align*}
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program solution |
\[ y = x \ln \left (x \right )+7 x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\ln \left (x \right )+7\right ) x \] |
Problem 63
| ODE |
\[ \boxed {-3 y+2 y^{\prime } x=9 x^{3}} \] |
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program solution |
\[ y = \frac {x^{\frac {3}{2}} c_{1}}{2}+3 x^{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 3 x^{3}+x^{\frac {3}{2}} c_{1} \] |
Problem 64
| ODE |
\[ \boxed {y+y^{\prime } x -3 y x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
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program solution |
N/A |
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Maple solution | \[ y \left (x \right ) = 0 \] |
Problem 65
| ODE |
\[ \boxed {3 y+y^{\prime } x=2 x^{5}} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {x^{8}-224}{4 x^{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{8}-224}{4 x^{3}} \] |
Problem 66
| ODE |
\[ \boxed {y^{\prime }+y={\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2} \] |
Problem 67
| ODE |
\[ \boxed {-3 y+y^{\prime } x=x^{3}} \] With initial conditions \begin {align*} [y \left (1\right ) = 10] \end {align*}
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program solution |
\[ y = \ln \left (x \right ) x^{3}+10 x^{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\ln \left (x \right )+10\right ) x^{3} \] |
Problem 68
| ODE |
\[ \boxed {2 y x +y^{\prime }=x} \] With initial conditions \begin {align*} [y \left (0\right ) = -2] \end {align*}
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program solution |
\[ y = \frac {1}{2}-\frac {5 \,{\mathrm e}^{-x^{2}}}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{2}-\frac {5 \,{\mathrm e}^{-x^{2}}}{2} \] |
Problem 69
| ODE |
\[ \boxed {y^{\prime }-\cos \left (x \right ) \left (1-y\right )=0} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 2] \end {align*}
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program solution |
\[ y = 1+{\mathrm e}^{-\sin \left (x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 1+{\mathrm e}^{-\sin \left (x \right )} \] |
Problem 70
| ODE |
\[ \boxed {y+\left (x +1\right ) y^{\prime }=\cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {\sin \left (x \right )+1}{x +1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sin \left (x \right )+1}{x +1} \] |
Problem 71
| ODE |
\[ \boxed {y^{\prime } x -2 y=x^{3} \cos \left (x \right )} \] |
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program solution |
\[ y = x^{2} \left (\sin \left (x \right )+c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\sin \left (x \right )+c_{1} \right ) x^{2} \] |
Problem 72
| ODE |
\[ \boxed {y \cot \left (x \right )+y^{\prime }=\cos \left (x \right )} \] |
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program solution |
\[ y = \frac {\sin \left (x \right )^{2}+2 c_{1}}{2 \sin \left (x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (2 \cos \left (x \right )^{2}-4 c_{1} -1\right ) \csc \left (x \right )}{4} \] |
Problem 73
| ODE |
\[ \boxed {y^{\prime }-y-y x=x +1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = {\mathrm e}^{\frac {1}{2} x^{2}+x}-1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -1+{\mathrm e}^{\frac {x \left (2+x \right )}{2}} \] |
Problem 74
| ODE |
\[ \boxed {-3 y+y^{\prime } x=x^{4} \cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (2 \pi \right ) = 0] \end {align*}
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program solution |
\[ y = x^{3} \sin \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sin \left (x \right ) x^{3} \] |
Problem 75
| ODE |
\[ \boxed {-2 y x +y^{\prime }=3 x^{2} {\mathrm e}^{x^{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 5] \end {align*}
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program solution |
\[ y = {\mathrm e}^{x^{2}} x^{3}+5 \,{\mathrm e}^{x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (x^{3}+5\right ) {\mathrm e}^{x^{2}} \] |
Problem 76
| ODE |
\[ \boxed {\left (-3+2 x \right ) y+y^{\prime } x=4 x^{4}} \] |
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program solution |
\[ y = x^{3} \left (2 \,{\mathrm e}^{2 x}+c_{1} \right ) {\mathrm e}^{-2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{3} \left (2+{\mathrm e}^{-2 x} c_{1} \right ) \] |
Problem 77
| ODE |
\[ \boxed {3 y x +\left (x^{2}+4\right ) y^{\prime }=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {\left (x^{2}+4\right )^{\frac {3}{2}}+16}{3 \left (x^{2}+4\right )^{\frac {3}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{3}+\frac {16}{3 \left (x^{2}+4\right )^{\frac {3}{2}}} \] |
Problem 78
| ODE |
\[ \boxed {3 y x^{3}+\left (x^{2}+1\right ) y^{\prime }=6 x \,{\mathrm e}^{-\frac {3 x^{2}}{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = 6 \left (x^{2}+1\right )^{\frac {3}{2}} {\mathrm e}^{-\frac {3 x^{2}}{2}} \left (\int _{0}^{x}\frac {\textit {\_a}}{\left (\textit {\_a}^{2}+1\right )^{\frac {5}{2}}}d \textit {\_a} \right )+\left (x^{2}+1\right )^{\frac {3}{2}} {\mathrm e}^{-\frac {3 x^{2}}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (3 x^{2} \sqrt {x^{2}+1}+3 \sqrt {x^{2}+1}-2\right ) {\mathrm e}^{-\frac {3 x^{2}}{2}} \] |
Problem 79
| ODE |
\[ \boxed {\left (x +y\right ) y^{\prime }+y=x} \] |
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program solution |
\[ -\frac {x \left (x -2 y\right )}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-c_{1} x -\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*} |
Problem 80
| ODE |
\[ \boxed {2 x y y^{\prime }-y^{2}=x^{2}} \] |
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program solution |
\[ -x +\frac {y^{2}}{x} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {\left (c_{1} +x \right ) x} \\ y \left (x \right ) &= -\sqrt {\left (c_{1} +x \right ) x} \\ \end{align*} |
Problem 81
| ODE |
\[ \boxed {y^{\prime } x -y-2 \sqrt {y x}=0} \] |
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program solution |
\[ y = x \ln \left (x \right )^{2}-2 x \ln \left (x \right ) c_{1} +c_{1}^{2} x \] Verified OK.
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Maple solution | \[ -\frac {y \left (x \right )}{\sqrt {x y \left (x \right )}}+\ln \left (x \right )-c_{1} = 0 \] |
Problem 82
| ODE |
\[ \boxed {\left (x -y\right ) y^{\prime }-y=x} \] |
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program solution |
\[ -\frac {\ln \left (x^{2}+y^{2}\right )}{2}-\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
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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 83
| ODE |
\[ \boxed {x \left (x +y\right ) y^{\prime }-y \left (x -y\right )=0} \] |
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program solution |
\[ y = \frac {x}{\operatorname {LambertW}\left (x^{2} {\mathrm e}^{-c_{1}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x}{\operatorname {LambertW}\left (c_{1} x^{2}\right )} \] |
Problem 84
| ODE |
\[ \boxed {\left (x +2 y\right ) y^{\prime }-y=0} \] |
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program solution |
\[ y = {\mathrm e}^{\operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )+\frac {c_{1}}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x}{2 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{2}}}{2}\right )} \] |
Problem 85
| ODE |
\[ \boxed {x y^{2} y^{\prime }-y^{3}=x^{3}} \] |
|
program solution |
\[ \frac {y^{3}}{3 x^{3}}-\ln \left (x \right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \left (3 \ln \left (x \right )+c_{1} \right )^{\frac {1}{3}} x \\ y \left (x \right ) &= -\frac {\left (3 \ln \left (x \right )+c_{1} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (3 \ln \left (x \right )+c_{1} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) x}{2} \\ \end{align*} |
Problem 86
| ODE |
\[ \boxed {y^{\prime } x^{2}-{\mathrm e}^{\frac {y}{x}} x^{2}-y x=0} \] |
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program solution |
\[ y = -\ln \left (\ln \left (-\frac {1}{c_{1} x}\right )\right ) x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \ln \left (-\frac {1}{\ln \left (x \right )+c_{1}}\right ) x \] |
Problem 87
| ODE |
\[ \boxed {y^{\prime } x^{2}-y x -y^{2}=0} \] |
|
program solution |
\[ y = -\frac {x}{\ln \left (x \right )+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x}{c_{1} -\ln \left (x \right )} \] |
Problem 88
| ODE |
\[ \boxed {x y y^{\prime }-3 y^{2}=x^{2}} \] |
|
program solution |
\[ \frac {2 y^{2}+x^{2}}{4 x^{6}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {4 c_{1} x^{4}-2}\, x}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 c_{1} x^{4}-2}\, x}{2} \\ \end{align*} |
Problem 89
| ODE |
\[ \boxed {\left (x^{2}-y^{2}\right ) y^{\prime }-2 y x=0} \] |
|
program solution |
\[ -\frac {x^{2}}{y}-y = c_{1} \] Verified OK.
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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 90
| ODE |
\[ \boxed {x y y^{\prime }-y^{2}-x \sqrt {4 x^{2}+y^{2}}=0} \] |
|
program solution |
\[ -\frac {\sqrt {4 x^{2}+y^{2}}}{x} = -\ln \left (x \right )+c_{1} \] Verified OK.
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Maple solution | \[ \frac {x \ln \left (x \right )-c_{1} x -\sqrt {4 x^{2}+y \left (x \right )^{2}}}{x} = 0 \] |
Problem 91
| ODE |
\[ \boxed {y^{\prime } x -y-\sqrt {x^{2}+y^{2}}=0} \] |
|
program solution |
\[ y = -\frac {{\mathrm e}^{-c_{1}} \left ({\mathrm e}^{2 c_{1}}-x^{2}\right )}{2} \] Verified OK.
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|
Maple solution | \[ \frac {-c_{1} x^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}+y \left (x \right )}{x^{2}} = 0 \] |
Problem 92
| ODE |
\[ \boxed {y y^{\prime }-\sqrt {x^{2}+y^{2}}=-x} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 \,{\mathrm e}^{c_{1}}+2 x \right )}{2}+\frac {c_{1}}{2}} \] Verified OK.
|
|
Maple solution | \[ \frac {-y \left (x \right )^{2} c_{1} +\sqrt {x^{2}+y \left (x \right )^{2}}+x}{y \left (x \right )^{2}} = 0 \] |
Problem 93
| ODE |
\[ \boxed {y \left (3 x +y\right )+x \left (x +y\right ) y^{\prime }=0} \] |
|
program solution |
\[ \frac {y x^{2} \left (y+2 x \right )}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {-c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}}{c_{1} x} \\ y \left (x \right ) &= \frac {-c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}}{c_{1} x} \\ \end{align*} |
Problem 94
| ODE |
\[ \boxed {y^{\prime }-\sqrt {x +y+1}=0} \] |
|
program solution |
\[ y = {\mathrm e}^{-2 \operatorname {LambertW}\left (-{\mathrm e}^{-\frac {x}{2}-1+\frac {c_{1}}{2}}\right )-x -2+c_{1}}-2 \,{\mathrm e}^{-\operatorname {LambertW}\left (-{\mathrm e}^{-\frac {x}{2}-1+\frac {c_{1}}{2}}\right )-\frac {x}{2}-1+\frac {c_{1}}{2}}-x \] Verified OK.
|
|
Maple solution | \[ x -2 \sqrt {1+x +y \left (x \right )}-\ln \left (-1+\sqrt {1+x +y \left (x \right )}\right )+\ln \left (1+\sqrt {1+x +y \left (x \right )}\right )+\ln \left (x +y \left (x \right )\right )-c_{1} = 0 \] |
Problem 95
| ODE |
\[ \boxed {y^{\prime }-\left (4 x +y\right )^{2}=0} \] |
|
program solution |
\[ y = \frac {\left (-4 c_{3} x -2\right ) \cos \left (2 x \right )-4 \left (x -\frac {c_{3}}{2}\right ) \sin \left (2 x \right )}{c_{3} \cos \left (2 x \right )+\sin \left (2 x \right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -4 x -2 \tan \left (-2 x +2 c_{1} \right ) \] |
Problem 96
| ODE |
\[ \boxed {\left (x +y\right ) y^{\prime }=0} \] |
|
program solution |
\[ y = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= c_{1} \\ \end{align*} |
Problem 97
| ODE |
\[ \boxed {2 y x +y^{\prime } x^{2}-5 y^{3}=0} \] |
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program solution |
\[ y = \frac {\sqrt {\left (c_{1} x^{5}+2\right ) x}}{c_{1} x^{5}+2} \] Verified OK. \[ y = -\frac {\sqrt {\left (c_{1} x^{5}+2\right ) x}}{c_{1} x^{5}+2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {\sqrt {\left (c_{1} x^{5}+2\right ) x}}{c_{1} x^{5}+2} \\ y \left (x \right ) &= -\frac {\sqrt {\left (c_{1} x^{5}+2\right ) x}}{c_{1} x^{5}+2} \\ \end{align*} |
Problem 98
| ODE |
\[ \boxed {2 x y^{3}+y^{2} y^{\prime }=6 x} \] |
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program solution |
\[ -\frac {x^{2}}{2}-\frac {\ln \left (y^{3}-3\right )}{6} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \left ({\mathrm e}^{-3 x^{2}} c_{1} +3\right )^{\frac {1}{3}} \\ y \left (x \right ) &= -\frac {\left ({\mathrm e}^{-3 x^{2}} c_{1} +3\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left ({\mathrm e}^{-3 x^{2}} c_{1} +3\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*} |
Problem 99
| ODE |
\[ \boxed {y^{\prime }-y^{3}-y=0} \] |
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program solution |
\[ y = \frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \] Verified OK. \[ y = -\frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {{\mathrm e}^{-2 x} c_{1} -1}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {{\mathrm e}^{-2 x} c_{1} -1}} \\ \end{align*} |
Problem 100
| ODE |
\[ \boxed {2 y x +y^{\prime } x^{2}-5 y^{4}=0} \] |
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program solution |
\[ y = \frac {7^{\frac {1}{3}} {\left (x \left (7 c_{1} x^{7}+15\right )^{2}\right )}^{\frac {1}{3}}}{7 c_{1} x^{7}+15} \] Verified OK. \[ y = \frac {7^{\frac {1}{3}} {\left (x \left (7 c_{1} x^{7}+15\right )^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{14 c_{1} x^{7}+30} \] Verified OK. \[ y = -\frac {7^{\frac {1}{3}} {\left (x \left (7 c_{1} x^{7}+15\right )^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{14 c_{1} x^{7}+30} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {7^{\frac {1}{3}} {\left (x \left (7 c_{1} x^{7}+15\right )^{2}\right )}^{\frac {1}{3}}}{7 c_{1} x^{7}+15} \\ y \left (x \right ) &= -\frac {7^{\frac {1}{3}} {\left (x \left (7 c_{1} x^{7}+15\right )^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{14 c_{1} x^{7}+30} \\ y \left (x \right ) &= \frac {7^{\frac {1}{3}} {\left (x \left (7 c_{1} x^{7}+15\right )^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{14 c_{1} x^{7}+30} \\ \end{align*} |