2.17.53 Problems 5201 to 5300

Problem 5201

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

Problem 5202

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

Problem 5203

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

Problem 5204

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

Problem 5205

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

Problem 5206

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-3 y=\sin \left (2 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {4 \left (\left (\cos \left (2 x \right )+\frac {7 \sin \left (2 x \right )}{4}\right ) {\mathrm e}^{3 x}-\frac {13 \,{\mathrm e}^{4 x}}{8}+\frac {5}{8}\right ) {\mathrm e}^{-3 x}}{65} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {4 \,{\mathrm e}^{-3 x} \left (\left (\cos \left (2 x \right )+\frac {7 \sin \left (2 x \right )}{4}\right ) {\mathrm e}^{3 x}-\frac {13 \,{\mathrm e}^{4 x}}{8}+\frac {5}{8}\right )}{65} \]

Problem 5207

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

Problem 5208

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -3] \end {align*}
program solution	\[ y = -\frac {2 \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )-6 \cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {2 \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right )-6 \cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {x}{2}}}{3} \]

Problem 5209

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=3 \,{\mathrm e}^{-2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {3 \,{\mathrm e}^{-2 x}}{5}+\frac {{\mathrm e}^{-x} \left (4 \cos \left (2 x \right )+13 \sin \left (2 x \right )\right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 \,{\mathrm e}^{-2 x}}{5}+\frac {{\mathrm e}^{-x} \left (4 \cos \left (2 x \right )+13 \sin \left (2 x \right )\right )}{10} \]

Problem 5210

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }-3 y=\operatorname {Heaviside}\left (x -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & x <4 \\ \frac {5 \sinh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) \sqrt {37}\, {\mathrm e}^{10-\frac {5 x}{2}}}{111}+\frac {\cosh \left (\frac {\sqrt {37}\, \left (x -4\right )}{2}\right ) {\mathrm e}^{10-\frac {5 x}{2}}}{3}-\frac {1}{3} & 4\le x \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {Heaviside}\left (x -4\right ) \left (-1+\frac {5 \sqrt {37}\, \sinh \left (\frac {\left (x -4\right ) \sqrt {37}}{2}\right ) {\mathrm e}^{-\frac {5 x}{2}+10}}{37}+\cosh \left (\frac {\left (x -4\right ) \sqrt {37}}{2}\right ) {\mathrm e}^{-\frac {5 x}{2}+10}\right )}{3} \]

Problem 5211

ODE	\[ \boxed {y^{\prime \prime \prime }-y=5} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -5+\frac {10 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}+\frac {5 \,{\mathrm e}^{x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -5+\frac {5 \,{\mathrm e}^{x}}{3}+\frac {10 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3} \]

Problem 5212

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

Problem 5213

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y=x^{2} {\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2, y^{\prime \prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{x} \left (x^{5}+60 x +60\right )}{60} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x} \left (x^{5}+60 x +60\right )}{60} \]

Problem 5214

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

Problem 5215

ODE	\[ \boxed {q^{\prime \prime }+9 q^{\prime }+14 q=\frac {\sin \left (t \right )}{2}} \] With initial conditions \begin {align*} [q \left (0\right ) = 0, q^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ q = -\frac {101 \,{\mathrm e}^{-7 t}}{500}+\frac {11 \,{\mathrm e}^{-2 t}}{50}-\frac {9 \cos \left (t \right )}{500}+\frac {13 \sin \left (t \right )}{500} \] Verified OK.
Maple solution	\[ q \left (t \right ) = -\frac {9 \cos \left (t \right )}{500}+\frac {13 \sin \left (t \right )}{500}-\frac {101 \,{\mathrm e}^{-7 t}}{500}+\frac {11 \,{\mathrm e}^{-2 t}}{50} \]

Problem 5216

ODE	\[ \boxed {\left (x +1\right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{3}}{9}+\frac {x^{4}}{24}-\frac {x^{5}}{50}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x^{3}}{9}+\frac {x^{4}}{24}-\frac {x^{5}}{50}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x +\frac {2 x^{3}}{27}-\frac {11 x^{4}}{144}+\frac {33 x^{5}}{1000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{9} x^{3}+\frac {1}{24} x^{4}-\frac {1}{50} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\frac {2}{27} x^{3}-\frac {11}{144} x^{4}+\frac {33}{1000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 5217

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

Problem 5218

ODE	\[ \boxed {y^{\prime \prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{3}}{6}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5219

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x -2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}+\frac {1}{2} x^{4}+\frac {1}{6} x^{6}\right ) y \left (0\right )+\left (x +\frac {2}{3} x^{3}+\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}+\frac {1}{2} x^{4}\right ) c_{1} +\left (x +\frac {2}{3} x^{3}+\frac {4}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}+\frac {1}{2} x^{4}\right ) y \left (0\right )+\left (x +\frac {2}{3} x^{3}+\frac {4}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5220

ODE	\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }+2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{18} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{3}}{3}\right ) c_{1} +\left (x -\frac {1}{4} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{3}}{3}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5221

ODE	\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{20} x^{5}+\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {7}{360} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{20} x^{5}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5222

ODE	\[ \boxed {y^{\prime \prime }+2 y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{4}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4}}{6}\right ) c_{1} +\left (x -\frac {1}{10} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{4}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{10} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5223

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}-\frac {1}{16} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}\right ) c_{1} +c_{2} x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{8} x^{4}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x +O\left (x^{6}\right ) \]

Problem 5224

ODE	\[ \boxed {y^{\prime \prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {x^{3}}{6}\right ) c_{1} +\left (x +\frac {1}{12} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x +\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 5225

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +y x^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1-x -\frac {x^{3}}{3}-\frac {x^{4}}{12}-\frac {x^{5}}{20}-\frac {x^{6}}{45}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1-\frac {x^{4}}{12}-x -\frac {x^{3}}{3}-\frac {x^{5}}{20}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-x -\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 5226

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

Problem 5227

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

Problem 5228

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

Problem 5229

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

Problem 5230

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

Problem 5231

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

Problem 5232

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

Problem 5233

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

Problem 5234

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

Problem 5235

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

Problem 5236

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

Problem 5237

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

Problem 5238

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

Problem 5239

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

Problem 5240

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

Problem 5241

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

Problem 5242

ODE	\[ \boxed {y^{2} \left (x^{2}+2\right )+\left (x^{3}+y^{3}\right ) \left (-y^{\prime } x +y\right )=0} \]
program solution	\[ \frac {\left (\frac {y^{3}}{x^{3}}-2\right ) x}{2 y}+\frac {1}{x^{2}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {6 \ln \left (x \right ) x^{2}+6 c_{1} x^{2}+\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}-6}{3 \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (c_{1} x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}}{6}+\left (-1-i \sqrt {3}\right ) \left (c_{1} x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 5243

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

Problem 5244

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

Problem 5245

ODE	\[ \boxed {2 y-y^{\prime } \left (4-x \right )=-1} \]
program solution	\[ y = \frac {-x^{2}+{\mathrm e}^{-2 c_{1}}+8 x -16}{2 x^{2}-16 x +32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 x -\frac {1}{2} x^{2}+c_{1}}{\left (x -4\right )^{2}} \]

Problem 5246

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

Problem 5247

ODE	\[ \boxed {2 y x +\left (2 x +3 y\right ) y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 5248

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

Problem 5249

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

Problem 5250

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

Problem 5251

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

Problem 5252

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

Problem 5253

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

Problem 5254

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

Problem 5255

ODE	\[ \boxed {2 y-\left (3 x +2 y-1\right ) y^{\prime }=-3 x -1} \]
program solution	\[ y = -\frac {3 x}{2}-\frac {2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {25 x}{4}+\frac {1}{4}-\frac {25 c_{1}}{2}}}{4}\right )}{5}+\frac {1}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 x}{2}-\frac {2 \operatorname {LambertW}\left (-\frac {c_{1} {\mathrm e}^{\frac {1}{4}-\frac {25 x}{4}}}{4}\right )}{5}+\frac {1}{10} \]

Problem 5256

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

Problem 5257

ODE	\[ \boxed {y^{2}+x y y^{\prime }=-x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
program solution	\[ \frac {x^{4}}{4}+\frac {x^{2} y^{2}}{2} = {\frac {3}{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\sqrt {-2 x^{4}+6}}{2 x} \]

Problem 5258

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

Problem 5259

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

Problem 5260

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

Problem 5261

ODE	\[ \boxed {-2 \sin \left (y\right )+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime }=-x -3} \]
program solution
Maple solution	\[ y \left (x \right ) = \arcsin \left (\frac {9 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-\frac {1}{3}-\frac {8 x}{9}}}{9}\right )}{8}+\frac {3}{8}+\frac {x}{2}\right ) \]

Problem 5262

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

Problem 5263

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

Problem 5264

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

Problem 5265

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

Problem 5266

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

Problem 5267

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

Problem 5268

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

Problem 5269

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

Problem 5270

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

Problem 5271

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

Problem 5272

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

Problem 5273

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

Problem 5274

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

Problem 5275

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

Problem 5276

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

Problem 5277

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

Problem 5278

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

Problem 5279

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

Problem 5280

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

Problem 5281

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

Problem 5282

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

Problem 5283

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

Problem 5284

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

Problem 5285

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

Problem 5286

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

Problem 5287

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

Problem 5288

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

Problem 5289

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

Problem 5290

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

Problem 5291

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

Problem 5292

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

Problem 5293

ODE	\[ \boxed {3 x^{2} y^{2}+4 \left (y x^{3}-3\right ) y^{\prime }=0} \]
program solution	\[ y^{4} x^{3}-4 y^{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{12} c_{1} +4 \textit {\_Z}^{3} c_{1} -x^{3}\right )^{9}+4}{x^{3}} \]

Problem 5294

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

Problem 5295

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

Problem 5296

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

Problem 5297

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

Problem 5298

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

Problem 5299

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

Problem 5300

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