2.17.126 Problems 12501 to 12600

Problem 12501

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

Problem 12502

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

Problem 12503

ODE	\[ \boxed {y^{\prime \prime }+12 y-7 y^{\prime }=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{4 x} \]

Problem 12504

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 12505

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

Problem 12506

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {\left (3+\sqrt {17}\right ) x}{2}}+\frac {c_{2} \sqrt {17}\, {\mathrm e}^{\frac {\left (-3+\sqrt {17}\right ) x}{2}}}{17} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\left (-3+\sqrt {17}\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {\left (3+\sqrt {17}\right ) x}{2}} \]

Problem 12507

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

Problem 12508

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{-\frac {x}{2}} \sqrt {3}}{3} \] 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 12509

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

Problem 12510

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

Problem 12511

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

Problem 12512

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

Problem 12513

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+9 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (1-i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (-1-i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (-1+i \sqrt {2}\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{x}+c_{4} {\mathrm e}^{-x}\right ) \cos \left (x \sqrt {2}\right )+\sin \left (x \sqrt {2}\right ) \left (c_{1} {\mathrm e}^{x}+c_{3} {\mathrm e}^{-x}\right ) \]

Problem 12514

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+x \,{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{2 x} c_{3} +x \,{\mathrm e}^{2 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{-2 x}+{\mathrm e}^{2 x} \left (c_{2} x +c_{1} \right ) \]

Problem 12515

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

Problem 12516

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

Problem 12517

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+12 y=x} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{4 x}+\frac {x}{12}+\frac {7}{144} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{3 x}+c_{1} {\mathrm e}^{4 x}+\frac {x}{12}+\frac {7}{144} \]

Problem 12518

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

Problem 12519

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=8 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{x}}{3}-\frac {2 \cos \left (2 x \right )}{5}-\frac {6 \sin \left (2 x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {2 \left (-\cos \left (2 x \right )-3 \sin \left (2 x \right )\right ) {\mathrm e}^{2 x}}{5}+c_{2} {\mathrm e}^{3 x}+c_{1} \right ) {\mathrm e}^{-2 x} \]

Problem 12520

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

Problem 12521

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

Problem 12522

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+5 y={\mathrm e}^{2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 x}+\frac {c_{2} {\mathrm e}^{-x}}{4}+\frac {{\mathrm e}^{2 x}}{21} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{7 x}+21 c_{2} {\mathrm e}^{4 x}+21 c_{1} \right ) {\mathrm e}^{-5 x}}{21} \]

Problem 12523

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

Problem 12524

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

Problem 12525

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+3 y={\mathrm e}^{-x} \cos \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (\sqrt {2}\, x \right )+\frac {c_{2} \sin \left (\sqrt {2}\, x \right ) {\mathrm e}^{x} \sqrt {2}}{2}+\frac {5 \,{\mathrm e}^{-x} \cos \left (x \right )}{41}-\frac {4 \,{\mathrm e}^{-x} \sin \left (x \right )}{41} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \cos \left (x \sqrt {2}\right ) c_{1} +{\mathrm e}^{x} \sin \left (x \sqrt {2}\right ) c_{2} +\frac {5 \,{\mathrm e}^{-x} \left (\cos \left (x \right )-\frac {4 \sin \left (x \right )}{5}\right )}{41} \]

Problem 12526

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

Problem 12527

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

Problem 12528

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

Problem 12529

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

Problem 12530

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

Problem 12531

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

Problem 12532

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+6 y=\sin \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{6 x}}{5}+\frac {7 \cos \left (x \right )}{74}+\frac {5 \sin \left (x \right )}{74} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{6 x} c_{2} +c_{1} {\mathrm e}^{x}+\frac {7 \cos \left (x \right )}{74}+\frac {5 \sin \left (x \right )}{74} \]

Problem 12533

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

Problem 12534

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

Problem 12535

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

Problem 12536

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )+x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 1] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\sin \left (t \right )+\cos \left (t \right ) \\ y \left (t \right ) &= \cos \left (t \right ) \\ \end{align*}

Problem 12537

ODE	\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+\cos \left (t \right )\\ y^{\prime }\left (t \right )&=-4 y \left (t \right )+4 \cos \left (t \right )+3 x \left (t \right )-\sin \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-3 t}}{3}+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} +\cos \left (t \right ) \\ \end{align*}

Problem 12538

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

Problem 12539

ODE	\[ \boxed {\frac {x^{2} y^{\prime }}{\left (x -y\right )^{2}}-\frac {y^{2}}{\left (x -y\right )^{2}}=0} \]
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 12540

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

Problem 12541

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

Problem 12542

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

Problem 12543

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

Problem 12544

ODE	\[ \boxed {y^{\prime \prime }-4 y=\sin \left (2 x \right ) {\mathrm e}^{2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}-\frac {\cos \left (2 x \right ) {\mathrm e}^{2 x}}{10}-\frac {\sin \left (2 x \right ) {\mathrm e}^{2 x}}{20} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (20 c_{2} -2 \cos \left (2 x \right )-\sin \left (2 x \right )\right ) {\mathrm e}^{2 x}}{20}+{\mathrm e}^{-2 x} c_{1} \]

Problem 12545

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

Problem 12546

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

Problem 12547

ODE	\[ \boxed {3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime }=0} \]
program solution	\[ -\ln \left ({\mathrm e}^{x}-1\right )+\frac {\ln \left (\tan \left (y\right )\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\arctan \left (-\frac {2 c_{1} \left ({\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-1\right )}{-c_{1}^{2} {\mathrm e}^{6 x}+6 c_{1}^{2} {\mathrm e}^{5 x}-15 c_{1}^{2} {\mathrm e}^{4 x}+20 c_{1}^{2} {\mathrm e}^{3 x}-15 c_{1}^{2} {\mathrm e}^{2 x}+6 c_{1}^{2} {\mathrm e}^{x}-c_{1}^{2}-1}, \frac {c_{1}^{2} {\mathrm e}^{6 x}-6 c_{1}^{2} {\mathrm e}^{5 x}+15 c_{1}^{2} {\mathrm e}^{4 x}-20 c_{1}^{2} {\mathrm e}^{3 x}+15 c_{1}^{2} {\mathrm e}^{2 x}-6 c_{1}^{2} {\mathrm e}^{x}+c_{1}^{2}-1}{-c_{1}^{2} {\mathrm e}^{6 x}+6 c_{1}^{2} {\mathrm e}^{5 x}-15 c_{1}^{2} {\mathrm e}^{4 x}+20 c_{1}^{2} {\mathrm e}^{3 x}-15 c_{1}^{2} {\mathrm e}^{2 x}+6 c_{1}^{2} {\mathrm e}^{x}-c_{1}^{2}-1}\right )}{2} \]

Problem 12548

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=5 x \left (t \right )+6 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (\sin \left (\sqrt {11}\, t \right ) c_{1} +\cos \left (\sqrt {11}\, t \right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{4 t} \left (\sin \left (\sqrt {11}\, t \right ) \sqrt {11}\, c_{2} -\cos \left (\sqrt {11}\, t \right ) \sqrt {11}\, c_{1} -2 \sin \left (\sqrt {11}\, t \right ) c_{1} -2 \cos \left (\sqrt {11}\, t \right ) c_{2} \right )}{3} \\ \end{align*}

Problem 12549

ODE	\begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )-10 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-3 t} \left (c_{1} \sin \left (3 t \right )-3 c_{2} \sin \left (3 t \right )+3 c_{1} \cos \left (3 t \right )+c_{2} \cos \left (3 t \right )\right )}{10} \\ \end{align*}

Problem 12550

ODE	\begin {align*} x^{\prime }\left (t \right )&=12 x \left (t \right )+18 y \left (t \right )\\ y^{\prime }\left (t \right )&=-8 x \left (t \right )-12 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} t +c_{2} \\ y \left (t \right ) &= \frac {1}{18} c_{1} -\frac {2}{3} c_{1} t -\frac {2}{3} c_{2} \\ \end{align*}

Problem 12551

ODE	\[ \boxed {y^{\prime }-y^{2}=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {-2 \operatorname {AiryAi}\left (1, -x \right ) \pi 3^{\frac {5}{6}}+3 \operatorname {AiryAi}\left (1, -x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}+3 \operatorname {AiryBi}\left (1, -x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {1}{6}}+2 \operatorname {AiryBi}\left (1, -x \right ) \pi 3^{\frac {1}{3}}}{-2 \operatorname {AiryAi}\left (-x \right ) \pi 3^{\frac {5}{6}}+3 \operatorname {AiryAi}\left (-x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}+3 \operatorname {AiryBi}\left (-x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {1}{6}}+2 \operatorname {AiryBi}\left (-x \right ) \pi 3^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2 \,3^{\frac {5}{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}\right ) \operatorname {AiryAi}\left (1, -x \right )+\operatorname {AiryBi}\left (1, -x \right ) \left (3 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{\frac {1}{3}}\right )}{\left (-2 \,3^{\frac {5}{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}\right ) \operatorname {AiryAi}\left (-x \right )+\operatorname {AiryBi}\left (-x \right ) \left (3 \,3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{\frac {1}{3}}\right )} \]

Problem 12552

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

Problem 12553

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

Problem 12554

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )+x \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right ) \\ y \left (t \right ) &= -\frac {2 c_{1} \cos \left (2 t \right )}{5}+\frac {2 c_{2} \sin \left (2 t \right )}{5}+\frac {c_{1} \sin \left (2 t \right )}{5}+\frac {c_{2} \cos \left (2 t \right )}{5} \\ \end{align*}

Problem 12555

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}} \sqrt {13}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}} \sqrt {13}}{2}-\frac {3 c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {13}\right ) t}{2}}}{2}-\frac {3 c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {13}\right ) t}{2}}}{2} \\ \end{align*}

Problem 12556

ODE	\begin {align*} x^{\prime }\left (t \right )&=-4 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}+c_{2} {\mathrm e}^{-\left (\sqrt {7}+3\right ) t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t} \sqrt {7}}{2}-\frac {c_{2} {\mathrm e}^{-\left (\sqrt {7}+3\right ) t} \sqrt {7}}{2}+\frac {c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}}{2}+\frac {c_{2} {\mathrm e}^{-\left (\sqrt {7}+3\right ) t}}{2} \\ \end{align*}

Problem 12557

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}} \sqrt {17}}{4}-\frac {c_{2} {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} \sqrt {17}}{4}+\frac {c_{1} {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}}}{4}+\frac {c_{2} {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}}}{4} \\ \end{align*}

Problem 12558

ODE	\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {3 c_{1} {\mathrm e}^{t}}{2}+c_{2} {\mathrm e}^{2 t} \\ \end{align*}

Problem 12559

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=y \left (t \right )-x \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} -\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right )}{2} \\ \end{align*}

Problem 12560

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (c_{2} t +c_{1} -c_{2} \right ) \\ \end{align*}

Problem 12561

ODE	\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+x \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= 2 c_{2} {\mathrm e}^{-t}+c_{1} \\ \end{align*}

Problem 12562

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-3 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{t}}{2}+c_{1} {\mathrm e}^{-3 t} \\ \end{align*}

Problem 12563

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{t}}{2}+c_{1} {\mathrm e}^{3 t} \\ \end{align*}

Problem 12564

ODE	\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-4 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-2+\sqrt {2}\right ) t}+c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t} \\ y \left (t \right ) &= \left (2+\sqrt {2}\right ) c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t}+\left (2-\sqrt {2}\right ) c_{1} {\mathrm e}^{\left (-2+\sqrt {2}\right ) t} \\ \end{align*}

Problem 12565

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )\\ y^{\prime }\left (t \right )&=y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t} \\ \end{align*}

Problem 12566

ODE	\begin {align*} x^{\prime }\left (t \right )&=0\\ y^{\prime }\left (t \right )&=x \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} \\ y \left (t \right ) &= c_{2} t +c_{1} \\ \end{align*}

Problem 12567

ODE	\[ \boxed {x^{\prime \prime }+x-x^{3}=0} \]
program solution	\[ \int _{}^{x}\frac {2}{\sqrt {2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} +2}}d \textit {\_a} = t +c_{2} \] Verified OK. \[ \int _{}^{x}-\frac {2}{\sqrt {2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} +2}}d \textit {\_a} = t +c_{3} \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{2} \sqrt {2}\, \sqrt {\frac {1}{c_{2}^{2}+1}}\, \operatorname {JacobiSN}\left (\frac {\left (\sqrt {2}\, t +2 c_{1} \right ) \sqrt {2}\, \sqrt {\frac {1}{c_{2}^{2}+1}}}{2}, c_{2}\right ) \]

Problem 12568

ODE	\[ \boxed {x^{\prime \prime }+x+x^{3}=0} \]
program solution	\[ \int _{}^{x}\frac {2}{\sqrt {-2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} -2}}d \textit {\_a} = t +c_{2} \] Verified OK. \[ \int _{}^{x}-\frac {2}{\sqrt {-2 \textit {\_a}^{4}-4 \textit {\_a}^{2}+8 c_{1} -2}}d \textit {\_a} = t +c_{3} \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{2} \operatorname {JacobiSN}\left (\frac {\left (\sqrt {3}\, \sqrt {2}\, t +2 c_{1} \right ) \sqrt {2}\, \sqrt {-\frac {1}{c_{2}^{2}-3}}}{2}, \frac {i c_{2} \sqrt {3}}{3}\right ) \sqrt {2}\, \sqrt {-\frac {1}{c_{2}^{2}-3}} \]

Problem 12569

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x-x^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12570

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x+x^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12571

ODE	\[ \boxed {x^{\prime \prime }-\left (2 \cos \left (x\right )-1\right ) \sin \left (x\right )=0} \]
program solution	\[ \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = t +c_{2} \] Verified OK. \[ \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = t +c_{3} \] Verified OK.
Maple solution	\begin{align*} \int _{}^{x \left (t \right )}\frac {1}{\sqrt {2 \sin \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -t -c_{2} &= 0 \\ -\left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {2 \sin \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} \right )-t -c_{2} &= 0 \\ \end{align*}

Problem 12572

ODE	\begin {align*} x^{\prime }&=x-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=-y \left (t \right )+x \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right ) \\ y \left (t \right ) &= -\frac {2 c_{1} \cos \left (2 t \right )}{5}+\frac {2 c_{2} \sin \left (2 t \right )}{5}+\frac {c_{1} \sin \left (2 t \right )}{5}+\frac {c_{2} \cos \left (2 t \right )}{5} \\ \end{align*}

Problem 12573

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

Problem 12574

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

Problem 12575

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

Problem 12576

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

Problem 12577

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

Problem 12578

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

Problem 12579

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

Problem 12580

ODE	\[ \boxed {y^{\prime }-2 \sqrt {{| y|}}=0} \]
program solution	\[ \frac {\left (\left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0
Maple solution	\[ x +\left (\left \{\begin {array}{cc} \sqrt {-y \left (x \right )} & y \left (x \right )\le 0 \\ -\sqrt {y \left (x \right )} & 0

Problem 12581

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

Problem 12582

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

Problem 12583

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

Problem 12584

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

Problem 12585

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

Problem 12586

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }-10 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{5 x}}{7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{7 x}+c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 12587

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

Problem 12588

ODE	\[ \boxed {y^{\prime \prime \prime }-7 y^{\prime \prime }+12 y^{\prime }=0} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{4 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} {\mathrm e}^{3 x}+c_{3} {\mathrm e}^{4 x} \]

Problem 12589

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

Problem 12590

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

Problem 12591

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

Problem 12592

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

Problem 12593

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

Problem 12594

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

Problem 12595

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

Problem 12596

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

Problem 12597

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

Problem 12598

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

Problem 12599

ODE	\[ \boxed {y^{\prime }-x \sqrt {y}=0} \]
program solution	\[ y = \frac {1}{16} x^{4}+\frac {1}{4} c_{1} x^{2}+\frac {1}{4} c_{1}^{2} \] Verified OK.
Maple solution	\[ \sqrt {y \left (x \right )}-\frac {x^{2}}{4}-c_{1} = 0 \]

Problem 12600

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