2.17.46 Problems 4501 to 4600

Problem 4501

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

Problem 4502

ODE	\[ \boxed {\left (-x^{3}+1\right ) y^{\prime }-2 \left (x +1\right ) y-y^{\frac {5}{2}}=0} \]
program solution	\[ y = \frac {{\left (\left (x^{2}+x +1\right ) \left (4 c_{1} x^{2}-8 c_{1} x +4 c_{1} -3\right )^{2}\right )}^{\frac {2}{3}} 2^{\frac {1}{3}}}{8 \left (-\frac {3}{4}+c_{1} \left (-1+x \right )^{2}\right )^{2}} \] Verified OK. \[ y = -\frac {2^{\frac {1}{3}} {\left (\left (x^{2}+x +1\right ) \left (4 c_{1} x^{2}-8 c_{1} x +4 c_{1} -3\right )^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )^{2}}{32 \left (-\frac {3}{4}+c_{1} \left (-1+x \right )^{2}\right )^{2}} \] Verified OK. \[ y = -\frac {2^{\frac {1}{3}} {\left (\left (x^{2}+x +1\right ) \left (4 c_{1} x^{2}-8 c_{1} x +4 c_{1} -3\right )^{2}\right )}^{\frac {2}{3}} \left (i-\sqrt {3}\right )^{2}}{32 \left (-\frac {3}{4}+c_{1} \left (-1+x \right )^{2}\right )^{2}} \] Verified OK.
Maple solution	\[ -\frac {\left (x -1\right )^{2} c_{1}}{x^{2}+x +1}+\frac {1}{y \left (x \right )^{\frac {3}{2}}}+\frac {3}{4 x^{2}+4 x +4} = 0 \]

Problem 4503

ODE	\[ \boxed {\tan \left (\theta \right ) r^{\prime }-r=\tan \left (\theta \right )^{2}} \]
program solution	\[ r = \frac {\ln \left (\sec \left (\theta \right )+\tan \left (\theta \right )\right )+c_{1}}{\csc \left (\theta \right )} \] Verified OK.
Maple solution	\[ r \left (\theta \right ) = \left (\ln \left (\sec \left (\theta \right )+\tan \left (\theta \right )\right )+c_{1} \right ) \sin \left (\theta \right ) \]

Problem 4504

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

Problem 4505

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

Problem 4506

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

Problem 4507

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

Problem 4508

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

Problem 4509

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

Problem 4510

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

Problem 4511

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

Problem 4512

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

Problem 4513

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

Problem 4514

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

Problem 4515

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

Problem 4516

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

Problem 4517

ODE	\[ \boxed {\left (x -\cos \left (y\right )\right ) y^{\prime }+\tan \left (y\right )=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = \frac {\pi }{6}\right ] \end {align*}
program solution	\[ \frac {\sin \left (y\right ) \left (-\cos \left (y\right )+2 x \right )}{2}-\frac {y}{2} = \frac {1}{2}-\frac {\sqrt {3}}{8}-\frac {\pi }{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (24 \sin \left (\textit {\_Z} \right ) x -6 \sin \left (2 \textit {\_Z} \right )+2 \pi +3 \sqrt {3}-12 \textit {\_Z} -12\right ) \]

Problem 4518

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

Problem 4519

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

Problem 4520

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

Problem 4521

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

Problem 4522

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

Problem 4523

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

Problem 4524

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

Problem 4525

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

Problem 4526

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

Problem 4527

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

Problem 4528

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

Problem 4529

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

Problem 4530

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.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1-\sqrt {-4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {-4 x^{2} c_{1}^{2}+1}}{2 c_{1}} \\ \end{align*}

Problem 4531

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

Problem 4532

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

Problem 4533

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

Problem 4534

ODE	\[ \boxed {y^{\prime }-\left (x^{2}+2 y-1\right )^{\frac {2}{3}}=-x} \]
program solution	\[ y = \frac {4}{27} c_{1}^{3}+\frac {4}{9} c_{1}^{2} x +\frac {4}{9} c_{1} x^{2}+\frac {4}{27} x^{3}-\frac {1}{2} x^{2}+\frac {1}{2} \] Verified OK.
Maple solution	\[ x -\frac {3 \left (x^{2}+2 y \left (x \right )-1\right )^{\frac {1}{3}}}{2}-c_{1} = 0 \]

Problem 4535

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

Problem 4536

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

Problem 4537

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

Problem 4538

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

Problem 4539

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

Problem 4540

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

Problem 4541

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

Problem 4542

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

Problem 4543

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

Problem 4544

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

Problem 4545

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

Problem 4546

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

Problem 4547

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

Problem 4548

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

Problem 4549

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

Problem 4550

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

Problem 4551

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

Problem 4552

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

Problem 4553

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

Problem 4554

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

Problem 4555

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

Problem 4556

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

Problem 4557

ODE	\[ \boxed {\left (1+x^{2}+y^{2}\right ) y^{\prime }+2 y x=-x^{2}-3} \]
program solution	\[ \frac {x^{3}}{3}+y x^{2}+3 x +\frac {y^{3}}{3}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {2}{3}}-4 x^{2}-4}{2 \left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+1\right )}{\left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {1}{3}}}{4}+\frac {\left (1+i \sqrt {3}\right ) \left (x^{2}+1\right )}{\left (-4 x^{3}-12 c_{1} -36 x +4 \sqrt {5 x^{6}+6 c_{1} x^{3}+30 x^{4}+9 c_{1}^{2}+54 c_{1} x +93 x^{2}+4}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 4558

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

Problem 4559

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

Problem 4560

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

Problem 4561

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

Problem 4562

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 4563

ODE	\[ \boxed {2 x y y^{\prime }-y^{2}=-3 x^{2}} \]
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 {c_{1} x -3 x^{2}} \\ \end{align*}

Problem 4564

ODE	\[ \boxed {\left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4}=0} \]
program solution	\[ \frac {y^{3}+x^{3}}{x y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {12^{\frac {1}{3}} \left (x 12^{\frac {1}{3}} c_{1} +{\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {2}{3}}\right )}{6 c_{1} {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) c_{1} 2^{\frac {2}{3}} x \right ) 2^{\frac {2}{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} \left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) c_{1} 2^{\frac {2}{3}} x \right ) 2^{\frac {2}{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 4565

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

Problem 4566

ODE	\[ \boxed {\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (2 x +y\right )=0} \]
program solution	\[ \frac {x^{2} \left (4 x +3 y\right )}{3}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, \left (i \sqrt {3}-1\right ) x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}+4 c_{1} x^{2}-\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4 \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}

Problem 4567

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

Problem 4568

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

Problem 4569

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

Problem 4570

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

Problem 4571

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

Problem 4572

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

Problem 4573

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

Problem 4574

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	\[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-x} \]

Problem 4575

ODE	\[ \boxed {6 y^{\prime \prime }-11 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{2}}+\frac {6 c_{2} {\mathrm e}^{\frac {4 x}{3}}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {4 x}{3}}+c_{2} {\mathrm e}^{\frac {x}{2}} \]

Problem 4576

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

Problem 4577

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }-6 y=0} \]
program solution	\[ y = {\mathrm e}^{3 x} c_{1} +{\mathrm e}^{\left (-\sqrt {2}-2\right ) x} c_{2} +{\mathrm e}^{\left (\sqrt {2}-2\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{\left (-2+\sqrt {2}\right ) x}+c_{3} {\mathrm e}^{-\left (2+\sqrt {2}\right ) x} \]

Problem 4578

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

Problem 4579

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

Problem 4580

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

Problem 4581

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

Problem 4582

ODE	\[ \boxed {y^{\prime \prime }+4 k y^{\prime }-12 k^{2} y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 k x}-\frac {c_{2} {\mathrm e}^{-6 k x}}{8 k} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{8 k x}+c_{2} \right ) {\mathrm e}^{-6 k x} \]

Problem 4583

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

Problem 4584

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

Problem 4585

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

Problem 4586

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]

Problem 4587

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

Problem 4588

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

Problem 4589

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

Problem 4590

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-11 y^{\prime \prime }-12 y^{\prime }+36 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+x \,{\mathrm e}^{-3 x} c_{2} +{\mathrm e}^{2 x} c_{3} +x \,{\mathrm e}^{2 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} x +c_{1} \right ) {\mathrm e}^{5 x}+x c_{4} +c_{3} \right ) {\mathrm e}^{-3 x} \]

Problem 4591

ODE	\[ \boxed {36 y^{\prime \prime \prime \prime }-37 y^{\prime \prime }+4 y^{\prime }+5 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{-\frac {x}{3}} c_{2} +{\mathrm e}^{\frac {5 x}{6}} c_{3} +{\mathrm e}^{\frac {x}{2}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{3} {\mathrm e}^{\frac {11 x}{6}}+c_{1} {\mathrm e}^{\frac {3 x}{2}}+c_{2} {\mathrm e}^{\frac {2 x}{3}}+c_{4} \right ) {\mathrm e}^{-x} \]

Problem 4592

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+36 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (\sqrt {5}+i\right ) x} c_{1} +{\mathrm e}^{\left (-i-\sqrt {5}\right ) x} c_{2} +{\mathrm e}^{\left (i-\sqrt {5}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {5}-i\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\sqrt {5}\, x} \sin \left (x \right )-c_{2} {\mathrm e}^{-\sqrt {5}\, x} \sin \left (x \right )+c_{3} {\mathrm e}^{\sqrt {5}\, x} \cos \left (x \right )+c_{4} {\mathrm e}^{-\sqrt {5}\, x} \cos \left (x \right ) \]

Problem 4593

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

Problem 4594

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

Problem 4595

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

Problem 4596

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+20 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x} \cos \left (4 x \right )+\frac {c_{2} {\mathrm e}^{2 x} \sin \left (4 x \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{1} \sin \left (4 x \right )+c_{2} \cos \left (4 x \right )\right ) \]

Problem 4597

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

Problem 4598

ODE	\[ \boxed {y^{\prime \prime \prime }+8 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right )+c_{3} {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )+c_{1} \right ) {\mathrm e}^{-2 x} \]

Problem 4599

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

Problem 4600

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