2.17.26 Problems 2501 to 2600

Problem 2501

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

Problem 2502

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

Problem 2503

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

Problem 2504

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

Problem 2505

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

Problem 2506

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

Problem 2507

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

Problem 2508

ODE	\[ \boxed {\left (5 x +y-7\right ) y^{\prime }-3 y=3 x +3} \]
program solution	\[ -\frac {\ln \left (y-3+3 x \right )}{2}+\frac {3 \ln \left (y+5-x \right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x -5\right ) \left (i \sqrt {3}-1\right ) \left (216 \sqrt {c_{1} \left (-2+x \right )^{2} \left (-\frac {1}{108}+\left (-2+x \right )^{2} c_{1} \right )}+1-216 \left (-2+x \right )^{2} c_{1} \right )^{\frac {2}{3}}+\left (-22 x +38\right ) \left (216 \sqrt {c_{1} \left (-2+x \right )^{2} \left (-\frac {1}{108}+\left (-2+x \right )^{2} c_{1} \right )}+1-216 \left (-2+x \right )^{2} c_{1} \right )^{\frac {1}{3}}-\left (1+i \sqrt {3}\right ) \left (x -5\right )}{i \sqrt {3}\, \left (216 \sqrt {c_{1} \left (-2+x \right )^{2} \left (-\frac {1}{108}+\left (-2+x \right )^{2} c_{1} \right )}+1-216 \left (-2+x \right )^{2} c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}-\left (216 \sqrt {c_{1} \left (-2+x \right )^{2} \left (-\frac {1}{108}+\left (-2+x \right )^{2} c_{1} \right )}+1-216 \left (-2+x \right )^{2} c_{1} \right )^{\frac {2}{3}}+2 \left (216 \sqrt {c_{1} \left (-2+x \right )^{2} \left (-\frac {1}{108}+\left (-2+x \right )^{2} c_{1} \right )}+1-216 \left (-2+x \right )^{2} c_{1} \right )^{\frac {1}{3}}-1} \]

Problem 2509

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

Problem 2510

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

Problem 2511

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

Problem 2512

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

Problem 2513

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

Problem 2514

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

Problem 2515

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

Problem 2516

ODE	\[ \boxed {f^{\prime \prime }+6 f^{\prime }+9 f={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [f \left (0\right ) = 0, f^{\prime }\left (0\right ) = \lambda ] \end {align*}
program solution	\[ f = \left (\lambda -\frac {1}{2}\right ) t \,{\mathrm e}^{-3 t}-\frac {{\mathrm e}^{-3 t}}{4}+\frac {{\mathrm e}^{-t}}{4} \] Verified OK.
Maple solution	\[ f \left (t \right ) = \frac {\left (-1+\left (4 \lambda -2\right ) t \right ) {\mathrm e}^{-3 t}}{4}+\frac {{\mathrm e}^{-t}}{4} \]

Problem 2517

ODE	\[ \boxed {f^{\prime \prime }+8 f^{\prime }+12 f=12 \,{\mathrm e}^{-4 t}} \] With initial conditions \begin {align*} [f \left (0\right ) = 0, f^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ f = \frac {3 \,{\mathrm e}^{-2 t}}{2}+\frac {3 \,{\mathrm e}^{-6 t}}{2}-3 \,{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ f \left (t \right ) = \frac {3 \,{\mathrm e}^{-2 t}}{2}+\frac {3 \,{\mathrm e}^{-6 t}}{2}-3 \,{\mathrm e}^{-4 t} \]

Problem 2518

ODE	\[ \boxed {f^{\prime \prime }+8 f^{\prime }+12 f=12 \,{\mathrm e}^{-4 t}} \] With initial conditions \begin {align*} [f \left (0\right ) = 0, f^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ f = {\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-6 t}-3 \,{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ f \left (t \right ) = {\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-6 t}-3 \,{\mathrm e}^{-4 t} \]

Problem 2519

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

Problem 2520

ODE	\[ \boxed {y^{\prime \prime \prime }-12 y^{\prime }+16 y=32 x -8} \]
program solution	\[ y = {\mathrm e}^{-4 x} \left (\left (c_{2} x +c_{1} \right ) {\mathrm e}^{6 x}+c_{3} \right )+1+2 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (2 x +1\right ) {\mathrm e}^{4 x}+\left (c_{3} x +c_{2} \right ) {\mathrm e}^{6 x}+c_{1} \right ) {\mathrm e}^{-4 x} \]

Problem 2521

ODE	\[ \boxed {0=-\frac {y^{\prime \prime }}{y}+\frac {{y^{\prime }}^{2}}{y^{2}}-\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y}+2 a^{2}} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {-x \,a^{2}+c_{1} \operatorname {arctanh}\left ({\mathrm e}^{2 a x}\right )-c_{2}}{a}} \sqrt {{\mathrm e}^{a x}-1}\, \sqrt {{\mathrm e}^{a x}+1}\, \sqrt {{\mathrm e}^{2 a x}+1} \]

Problem 2522

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

Problem 2523

ODE	\[ \boxed {\left (x +1\right )^{2} y^{\prime \prime }+3 \left (x +1\right ) y^{\prime }+y=x^{2}} \]
program solution	\[ y = \frac {\left (18 c_{1} -6\right ) \ln \left (x +1\right )+2 x^{3}-3 x^{2}+6 x +18 c_{2}}{18 x +18} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (18 c_{1} -6\right ) \ln \left (x +1\right )+2 x^{3}-3 x^{2}+6 x +18 c_{2}}{18 x +18} \]

Problem 2524

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

Problem 2525

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

Problem 2526

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

Problem 2527

ODE	Solve \begin {gather*} \boxed {2 y y^{\prime \prime \prime }+2 \left (y+3 y^{\prime }\right ) y^{\prime \prime }+2 \left (y^{\prime }\right )^{2}-\sin \left (x \right )=0} \end {gather*}
program solution	N/A
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-4 \left (\left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+c_{1} \left (x -1\right )+c_{3} \right ) {\mathrm e}^{x}-c_{2} \right ) {\mathrm e}^{x}}\, {\mathrm e}^{-x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-4 \left (\left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+c_{1} \left (x -1\right )+c_{3} \right ) {\mathrm e}^{x}-c_{2} \right ) {\mathrm e}^{x}}\, {\mathrm e}^{-x}}{2} \\ \end{align*}

Problem 2528

ODE	\[ \boxed {x y^{\prime \prime \prime }+2 y^{\prime \prime }=A x} \]
program solution	\[ y = c_{1} x +\frac {{\mathrm e}^{-c_{2}} \ln \left (x \right )}{c_{3}}+c_{4} +\frac {A \,x^{3}}{18} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {A \,x^{3}}{18}-\ln \left (x \right ) c_{1} +c_{2} x +c_{3} \]

Problem 2529

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+6\right ) y={\mathrm e}^{-x^{2}} \sin \left (2 x \right )} \]
program solution	\[ y = \frac {{\mathrm e}^{-x^{2}} \left (c_{2} \sin \left (2 x \right )+2 c_{1} \cos \left (2 x \right )\right )}{2}+\frac {{\mathrm e}^{-x^{2}} \left (\sin \left (2 x \right )-4 x \cos \left (2 x \right )\right )}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (x -4 c_{2} \right ) \cos \left (2 x \right )-4 \sin \left (2 x \right ) c_{1} \right ) {\mathrm e}^{-x^{2}}}{4} \]

Problem 2530

ODE	\[ \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-3 z y^{\prime }+\lambda y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = \left (1-\frac {1}{2} \lambda \,z^{2}+\frac {1}{24} \lambda ^{2} z^{4}-\frac {1}{3} \lambda \,z^{4}-\frac {1}{720} z^{6} \lambda ^{3}+\frac {2}{45} z^{6} \lambda ^{2}-\frac {4}{15} z^{6} \lambda \right ) y \left (0\right )+\left (z -\frac {1}{6} z^{3} \lambda +\frac {1}{2} z^{3}+\frac {1}{120} \lambda ^{2} z^{5}-\frac {3}{20} \lambda \,z^{5}+\frac {3}{8} z^{5}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {\lambda \,z^{2}}{2}+\left (\frac {1}{24} \lambda ^{2}-\frac {1}{3} \lambda \right ) z^{4}\right ) c_{1} +\left (z +\left (-\frac {\lambda }{6}+\frac {1}{2}\right ) z^{3}+\left (\frac {1}{120} \lambda ^{2}-\frac {3}{20} \lambda +\frac {3}{8}\right ) z^{5}\right ) c_{2} +O\left (z^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (1-\frac {\lambda \,z^{2}}{2}+\frac {\lambda \left (\lambda -8\right ) z^{4}}{24}\right ) y \left (0\right )+\left (z -\frac {\left (\lambda -3\right ) z^{3}}{6}+\frac {\left (\lambda -3\right ) \left (\lambda -15\right ) z^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

Problem 2531

ODE	\[ \boxed {4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} \sqrt {z}\, \left (1+\frac {z}{3}+\frac {z^{2}}{15}+\frac {z^{3}}{105}+\frac {z^{4}}{945}+\frac {z^{5}}{10395}+O\left (z^{6}\right )\right )+c_{2} \left (1+\frac {z}{2}+\frac {z^{2}}{8}+\frac {z^{3}}{48}+\frac {z^{4}}{384}+\frac {z^{5}}{3840}+O\left (z^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = c_{1} \sqrt {z}\, \left (1+\frac {1}{3} z +\frac {1}{15} z^{2}+\frac {1}{105} z^{3}+\frac {1}{945} z^{4}+\frac {1}{10395} z^{5}+\operatorname {O}\left (z^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} z +\frac {1}{8} z^{2}+\frac {1}{48} z^{3}+\frac {1}{384} z^{4}+\frac {1}{3840} z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \]

Problem 2532

ODE	\[ \boxed {z y^{\prime \prime }-2 y^{\prime }+9 z^{5} y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} z^{3} \left (1-\frac {z^{6}}{6}+O\left (z^{7}\right )\right )+c_{2} \left (1-\frac {z^{6}}{2}+O\left (z^{7}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = c_{1} z^{3} \left (1-\frac {1}{6} z^{6}+\operatorname {O}\left (z^{7}\right )\right )+c_{2} \left (12-6 z^{6}+\operatorname {O}\left (z^{7}\right )\right ) \]

Problem 2533

ODE	\[ \boxed {f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ f = \left (1-2 z^{2}-\frac {4}{3} z^{3}+\frac {2}{3} z^{4}+\frac {14}{15} z^{5}+\frac {2}{45} z^{6}\right ) f \left (0\right )+\left (z +z^{2}-\frac {1}{3} z^{3}-\frac {5}{6} z^{4}-\frac {1}{6} z^{5}+\frac {5}{18} z^{6}\right ) f^{\prime }\left (0\right )+O\left (z^{6}\right ) \] Verified OK. \[ f = \left (1-2 z^{2}-\frac {4}{3} z^{3}+\frac {2}{3} z^{4}+\frac {14}{15} z^{5}\right ) c_{1} +\left (z +z^{2}-\frac {1}{3} z^{3}-\frac {5}{6} z^{4}-\frac {1}{6} z^{5}\right ) c_{2} +O\left (z^{6}\right ) \] Verified OK.
Maple solution	\[ f \left (z \right ) = \left (1-2 z^{2}-\frac {4}{3} z^{3}+\frac {2}{3} z^{4}+\frac {14}{15} z^{5}\right ) f \left (0\right )+\left (z +z^{2}-\frac {1}{3} z^{3}-\frac {5}{6} z^{4}-\frac {1}{6} z^{5}\right ) D\left (f \right )\left (0\right )+O\left (z^{6}\right ) \]

Problem 2534

ODE	\[ \boxed {z^{2} y^{\prime \prime }-\frac {3 z y^{\prime }}{2}+\left (z +1\right ) y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} z^{2} \left (1-\frac {2 z}{5}+\frac {2 z^{2}}{35}-\frac {4 z^{3}}{945}+\frac {2 z^{4}}{10395}-\frac {4 z^{5}}{675675}+O\left (z^{6}\right )\right )+c_{2} \sqrt {z}\, \left (1+2 z -2 z^{2}+\frac {4 z^{3}}{9}-\frac {2 z^{4}}{45}+\frac {4 z^{5}}{1575}+O\left (z^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = c_{1} \sqrt {z}\, \left (1+2 z -2 z^{2}+\frac {4}{9} z^{3}-\frac {2}{45} z^{4}+\frac {4}{1575} z^{5}+\operatorname {O}\left (z^{6}\right )\right )+c_{2} z^{2} \left (1-\frac {2}{5} z +\frac {2}{35} z^{2}-\frac {4}{945} z^{3}+\frac {2}{10395} z^{4}-\frac {4}{675675} z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \]

Problem 2535

ODE	\[ \boxed {z y^{\prime \prime }-2 y^{\prime }+z y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} z^{3} \left (1-\frac {z^{2}}{10}+\frac {z^{4}}{280}+O\left (z^{6}\right )\right )+c_{2} \left (1+\frac {z^{2}}{2}-\frac {z^{4}}{8}+O\left (z^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = c_{1} z^{3} \left (1-\frac {1}{10} z^{2}+\frac {1}{280} z^{4}+\operatorname {O}\left (z^{6}\right )\right )+c_{2} \left (12+6 z^{2}-\frac {3}{2} z^{4}+\operatorname {O}\left (z^{6}\right )\right ) \]

Problem 2536

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

Problem 2537

ODE	\[ \boxed {z \left (1-z \right ) y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} \left (-\lambda z +1+\frac {\left (\lambda -1\right ) \lambda \,z^{2}}{4}-\frac {\left (\lambda -4\right ) \left (\lambda -1\right ) \lambda \,z^{3}}{36}+\frac {\left (\lambda -9\right ) \left (\lambda -4\right ) \left (\lambda -1\right ) \lambda \,z^{4}}{576}-\frac {\left (\lambda -16\right ) \left (\lambda -9\right ) \left (\lambda -4\right ) \left (\lambda -1\right ) \lambda \,z^{5}}{14400}+O\left (z^{6}\right )\right )+c_{2} \left (\left (-\lambda z +1+\frac {\left (\lambda -1\right ) \lambda \,z^{2}}{4}-\frac {\left (\lambda -4\right ) \left (\lambda -1\right ) \lambda \,z^{3}}{36}+\frac {\left (\lambda -9\right ) \left (\lambda -4\right ) \left (\lambda -1\right ) \lambda \,z^{4}}{576}-\frac {\left (\lambda -16\right ) \left (\lambda -9\right ) \left (\lambda -4\right ) \left (\lambda -1\right ) \lambda \,z^{5}}{14400}+O\left (z^{6}\right )\right ) \ln \left (z \right )+2 \lambda z +\left (-\frac {\lambda }{2}-\frac {3 \left (\lambda -1\right ) \lambda }{4}\right ) z^{2}+\left (-\frac {\left (-\lambda +1\right ) \lambda }{9}-\frac {\left (-\lambda +4\right ) \lambda }{18}+\frac {11 \left (-\lambda +4\right ) \left (-\lambda +1\right ) \lambda }{108}\right ) z^{3}+\left (-\frac {\left (\lambda -4\right ) \left (\lambda -1\right ) \lambda }{96}-\frac {\left (\lambda -9\right ) \left (\lambda -1\right ) \lambda }{144}-\frac {\left (\lambda -9\right ) \left (\lambda -4\right ) \lambda }{288}-\frac {25 \left (\lambda -9\right ) \left (\lambda -4\right ) \left (\lambda -1\right ) \lambda }{3456}\right ) z^{4}+\left (-\frac {\left (-\lambda +9\right ) \left (-\lambda +4\right ) \left (-\lambda +1\right ) \lambda }{1800}-\frac {\left (-\lambda +16\right ) \left (-\lambda +4\right ) \left (-\lambda +1\right ) \lambda }{2400}-\frac {\left (-\lambda +16\right ) \left (-\lambda +9\right ) \left (-\lambda +1\right ) \lambda }{3600}-\frac {\left (-\lambda +16\right ) \left (-\lambda +9\right ) \left (-\lambda +4\right ) \lambda }{7200}+\frac {137 \left (-\lambda +16\right ) \left (-\lambda +9\right ) \left (-\lambda +4\right ) \left (-\lambda +1\right ) \lambda }{432000}\right ) z^{5}+O\left (z^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (2 \lambda z +\left (\frac {1}{4} \lambda -\frac {3}{4} \lambda ^{2}\right ) z^{2}+\left (-\frac {37}{108} \lambda ^{2}+\frac {2}{27} \lambda +\frac {11}{108} \lambda ^{3}\right ) z^{3}+\left (\frac {139}{1728} \lambda ^{3}-\frac {649}{3456} \lambda ^{2}+\frac {1}{32} \lambda -\frac {25}{3456} \lambda ^{4}\right ) z^{4}+\left (-\frac {13}{1600} \lambda ^{4}+\frac {8467}{144000} \lambda ^{3}-\frac {2527}{21600} \lambda ^{2}+\frac {2}{125} \lambda +\frac {137}{432000} \lambda ^{5}\right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) c_{2} +\left (1-\lambda z +\frac {1}{4} \left (-1+\lambda \right ) \lambda z^{2}-\frac {1}{36} \lambda \left (\lambda ^{2}-5 \lambda +4\right ) z^{3}+\frac {1}{576} \lambda \left (\lambda ^{3}-14 \lambda ^{2}+49 \lambda -36\right ) z^{4}-\frac {1}{14400} \lambda \left (-1+\lambda \right ) \left (\lambda -4\right ) \left (\lambda -16\right ) \left (\lambda -9\right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \left (c_{2} \ln \left (z \right )+c_{1} \right ) \]

Problem 2538

ODE	\[ \boxed {z y^{\prime \prime }+\left (2 z -3\right ) y^{\prime }+\frac {4 y}{z}=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} z^{2} \left (6 z^{2}-4 z +1-\frac {16 z^{3}}{3}+\frac {10 z^{4}}{3}-\frac {8 z^{5}}{5}+O\left (z^{6}\right )\right )+c_{2} \left (z^{2} \left (6 z^{2}-4 z +1-\frac {16 z^{3}}{3}+\frac {10 z^{4}}{3}-\frac {8 z^{5}}{5}+O\left (z^{6}\right )\right ) \ln \left (z \right )+z^{2} \left (-13 z^{2}+6 z +\frac {124 z^{3}}{9}-\frac {173 z^{4}}{18}+\frac {374 z^{5}}{75}+O\left (z^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (\left (c_{2} \ln \left (z \right )+c_{1} \right ) \left (1-4 z +6 z^{2}-\frac {16}{3} z^{3}+\frac {10}{3} z^{4}-\frac {8}{5} z^{5}+\operatorname {O}\left (z^{6}\right )\right )+\left (6 z -13 z^{2}+\frac {124}{9} z^{3}-\frac {173}{18} z^{4}+\frac {374}{75} z^{5}+\operatorname {O}\left (z^{6}\right )\right ) c_{2} \right ) z^{2} \]

Problem 2539

ODE	\[ \boxed {\left (z^{2}+5 z +6\right ) y^{\prime \prime }+2 y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = \left (1-\frac {1}{6} z^{2}+\frac {5}{108} z^{3}-\frac {13}{1296} z^{4}+\frac {5}{2592} z^{5}-\frac {17}{58320} z^{6}\right ) y \left (0\right )+\left (z -\frac {1}{18} z^{3}+\frac {5}{216} z^{4}-\frac {17}{2160} z^{5}+\frac {5}{1944} z^{6}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} z^{2}+\frac {5}{108} z^{3}-\frac {13}{1296} z^{4}+\frac {5}{2592} z^{5}\right ) c_{1} +\left (z -\frac {1}{18} z^{3}+\frac {5}{216} z^{4}-\frac {17}{2160} z^{5}\right ) c_{2} +O\left (z^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (1-\frac {1}{6} z^{2}+\frac {5}{108} z^{3}-\frac {13}{1296} z^{4}+\frac {5}{2592} z^{5}\right ) y \left (0\right )+\left (z -\frac {1}{18} z^{3}+\frac {5}{216} z^{4}-\frac {17}{2160} z^{5}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

Problem 2540

ODE	\[ \boxed {\left (z^{2}+5 z +7\right ) y^{\prime \prime }+2 y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = \left (1-\frac {1}{7} z^{2}+\frac {5}{147} z^{3}-\frac {11}{2058} z^{4}+\frac {5}{14406} z^{5}+\frac {289}{1512630} z^{6}\right ) y \left (0\right )+\left (z -\frac {1}{21} z^{3}+\frac {5}{294} z^{4}-\frac {47}{10290} z^{5}+\frac {5}{4802} z^{6}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{7} z^{2}+\frac {5}{147} z^{3}-\frac {11}{2058} z^{4}+\frac {5}{14406} z^{5}\right ) c_{1} +\left (z -\frac {1}{21} z^{3}+\frac {5}{294} z^{4}-\frac {47}{10290} z^{5}\right ) c_{2} +O\left (z^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (1-\frac {1}{7} z^{2}+\frac {5}{147} z^{3}-\frac {11}{2058} z^{4}+\frac {5}{14406} z^{5}\right ) y \left (0\right )+\left (z -\frac {1}{21} z^{3}+\frac {5}{294} z^{4}-\frac {47}{10290} z^{5}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

Problem 2541

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

Problem 2542

ODE	\[ \boxed {z y^{\prime \prime }+\left (1-z \right ) y^{\prime }+\lambda y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = c_{1} \left (-\lambda z +1+\frac {\left (\lambda -1\right ) \lambda \,z^{2}}{4}-\frac {\left (\lambda -2\right ) \left (\lambda -1\right ) \lambda \,z^{3}}{36}+\frac {\left (\lambda -3\right ) \left (\lambda -2\right ) \left (\lambda -1\right ) \lambda \,z^{4}}{576}-\frac {\left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (\lambda -1\right ) \lambda \,z^{5}}{14400}+O\left (z^{6}\right )\right )+c_{2} \left (\left (-\lambda z +1+\frac {\left (\lambda -1\right ) \lambda \,z^{2}}{4}-\frac {\left (\lambda -2\right ) \left (\lambda -1\right ) \lambda \,z^{3}}{36}+\frac {\left (\lambda -3\right ) \left (\lambda -2\right ) \left (\lambda -1\right ) \lambda \,z^{4}}{576}-\frac {\left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (\lambda -1\right ) \lambda \,z^{5}}{14400}+O\left (z^{6}\right )\right ) \ln \left (z \right )+\left (1+2 \lambda \right ) z +\left (-\frac {\lambda }{2}+\frac {1}{4}-\frac {3 \left (\lambda -1\right ) \lambda }{4}\right ) z^{2}+\left (-\frac {\left (-\lambda +1\right ) \lambda }{36}-\frac {\left (-\lambda +2\right ) \lambda }{36}+\frac {\left (-\lambda +2\right ) \left (-\lambda +1\right )}{36}+\frac {11 \left (-\lambda +2\right ) \left (-\lambda +1\right ) \lambda }{108}\right ) z^{3}+\left (-\frac {\left (\lambda -2\right ) \left (\lambda -1\right ) \lambda }{576}-\frac {\left (\lambda -3\right ) \left (\lambda -1\right ) \lambda }{576}-\frac {\left (\lambda -3\right ) \left (\lambda -2\right ) \lambda }{576}-\frac {\left (\lambda -3\right ) \left (\lambda -2\right ) \left (\lambda -1\right )}{576}-\frac {25 \left (\lambda -3\right ) \left (\lambda -2\right ) \left (\lambda -1\right ) \lambda }{3456}\right ) z^{4}+\left (-\frac {\left (-\lambda +3\right ) \left (-\lambda +2\right ) \left (-\lambda +1\right ) \lambda }{14400}-\frac {\left (-\lambda +4\right ) \left (-\lambda +2\right ) \left (-\lambda +1\right ) \lambda }{14400}-\frac {\left (-\lambda +4\right ) \left (-\lambda +3\right ) \left (-\lambda +1\right ) \lambda }{14400}-\frac {\left (-\lambda +4\right ) \left (-\lambda +3\right ) \left (-\lambda +2\right ) \lambda }{14400}+\frac {\left (-\lambda +4\right ) \left (-\lambda +3\right ) \left (-\lambda +2\right ) \left (-\lambda +1\right )}{14400}+\frac {137 \left (-\lambda +4\right ) \left (-\lambda +3\right ) \left (-\lambda +2\right ) \left (-\lambda +1\right ) \lambda }{432000}\right ) z^{5}+O\left (z^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (\left (2 \lambda +1\right ) z +\left (\frac {1}{4} \lambda +\frac {1}{4}-\frac {3}{4} \lambda ^{2}\right ) z^{2}+\left (-\frac {2}{9} \lambda ^{2}+\frac {1}{27} \lambda +\frac {1}{18}+\frac {11}{108} \lambda ^{3}\right ) z^{3}+\left (\frac {7}{192} \lambda ^{3}-\frac {167}{3456} \lambda ^{2}+\frac {1}{192} \lambda +\frac {1}{96}-\frac {25}{3456} \lambda ^{4}\right ) z^{4}+\left (\frac {1}{1500} \lambda -\frac {37}{4320} \lambda ^{2}+\frac {719}{86400} \lambda ^{3}+\frac {1}{600}-\frac {61}{21600} \lambda ^{4}+\frac {137}{432000} \lambda ^{5}\right ) z^{5}+\operatorname {O}\left (z^{6}\right )\right ) c_{2} +\left (1-\lambda z +\frac {1}{4} \left (-1+\lambda \right ) \lambda z^{2}-\frac {1}{36} \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{3}+\frac {1}{576} \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{4}-\frac {1}{14400} \left (\lambda -4\right ) \left (\lambda -3\right ) \left (\lambda -2\right ) \left (-1+\lambda \right ) \lambda z^{5}+\operatorname {O}\left (z^{6}\right )\right ) \left (c_{2} \ln \left (z \right )+c_{1} \right ) \]

Problem 2543

ODE	\[ \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-z y^{\prime }+m^{2} y=0} \] With the expansion point for the power series method at \(z = 0\).
program solution	\[ y = \left (1-\frac {1}{2} m^{2} z^{2}+\frac {1}{24} m^{4} z^{4}-\frac {1}{6} m^{2} z^{4}-\frac {1}{720} z^{6} m^{6}+\frac {1}{36} z^{6} m^{4}-\frac {4}{45} z^{6} m^{2}\right ) y \left (0\right )+\left (z -\frac {1}{6} m^{2} z^{3}+\frac {1}{6} z^{3}+\frac {1}{120} m^{4} z^{5}-\frac {1}{12} m^{2} z^{5}+\frac {3}{40} z^{5}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {m^{2} z^{2}}{2}+\left (\frac {1}{24} m^{4}-\frac {1}{6} m^{2}\right ) z^{4}\right ) c_{1} +\left (z +\left (-\frac {m^{2}}{6}+\frac {1}{6}\right ) z^{3}+\left (\frac {1}{120} m^{4}-\frac {1}{12} m^{2}+\frac {3}{40}\right ) z^{5}\right ) c_{2} +O\left (z^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (z \right ) = \left (1-\frac {m^{2} z^{2}}{2}+\frac {m^{2} \left (m^{2}-4\right ) z^{4}}{24}\right ) y \left (0\right )+\left (z -\frac {\left (m^{2}-1\right ) z^{3}}{6}+\frac {\left (m^{4}-10 m^{2}+9\right ) z^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

Problem 2544

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 2545

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

Problem 2546

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

Problem 2547

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

Problem 2548

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

Problem 2549

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

Problem 2550

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

Problem 2551

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

Problem 2552

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

Problem 2553

ODE	\[ \boxed {y^{\prime }-\frac {y x^{2}-32}{-x^{2}+16}=32} \]
program solution	\[ y = -\frac {\left (1440 \,{\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-4-x \right ) x^{2}+11520 \,{\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-4-x \right ) x -32 x^{2} {\mathrm e}^{x}-c_{1} x^{2}+23040 \,{\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-4-x \right )+1696 x \,{\mathrm e}^{x}-8 c_{1} x +7168 \,{\mathrm e}^{x}-16 c_{1} \right ) {\mathrm e}^{-x}}{x^{2}-8 x +16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-1440 \,{\mathrm e}^{-4-x} \left (x +4\right )^{2} \operatorname {expIntegral}_{1}\left (-4-x \right )+c_{1} \left (x +4\right )^{2} {\mathrm e}^{-x}+32 x^{2}-1696 x -7168}{\left (x -4\right )^{2}} \]

Problem 2554

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

Problem 2555

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

Problem 2556

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

Problem 2557

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

Problem 2558

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

Problem 2559

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

Problem 2560

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

Problem 2561

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

Problem 2562

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

Problem 2563

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

Problem 2564

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

Problem 2565

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

Problem 2566

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

Problem 2567

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

Problem 2568

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

Problem 2569

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

Problem 2570

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

Problem 2571

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

Problem 2572

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

Problem 2573

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

Problem 2574

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

Problem 2575

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

Problem 2576

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

Problem 2577

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

Problem 2578

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

Problem 2579

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

Problem 2580

ODE	\[ \boxed {y^{\prime }-\frac {y^{2}+2 y x -2 x^{2}}{x^{2}-y x +y^{2}}=0} \]
program solution	\[ \frac {\ln \left (y+x \right )}{2}-\frac {\ln \left (y-x \right )}{2}+\ln \left (y-2 x \right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \left (-\operatorname {RootOf}\left (2 \textit {\_Z}^{6}+\left (9 c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-6 x^{2} c_{1} \textit {\_Z}^{2}+c_{1} x^{2}\right )^{2}+1\right )}{\operatorname {RootOf}\left (2 \textit {\_Z}^{6}+\left (9 c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-6 x^{2} c_{1} \textit {\_Z}^{2}+c_{1} x^{2}\right )^{2}} \]

Problem 2581

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

Problem 2582

ODE	\[ \boxed {y^{\prime } x^{2}-y^{2}-3 y x=x^{2}} \]
program solution	\[ y = -\frac {\left (\ln \left (x \right )+c_{3} +1\right ) x}{\ln \left (x \right )+c_{3}} \] 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 2583

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

Problem 2584

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

Problem 2585

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

Problem 2586

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

Problem 2587

ODE	\[ \boxed {y^{\prime \prime }-25 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 x}+\frac {c_{2} {\mathrm e}^{5 x}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{-5 x} \]

Problem 2588

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

Problem 2589

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

Problem 2590

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

Problem 2591

ODE	\[ \boxed {y^{\prime }-\frac {y}{2 x}=0} \]
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 2592

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

Problem 2593

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

Problem 2594

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

Problem 2595

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

Problem 2596

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +13 y=0} \]
program solution	\[ y = c_{1} x^{2-3 i}-\frac {i c_{2} x^{2+3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} \sin \left (3 \ln \left (x \right )\right )+c_{2} \cos \left (3 \ln \left (x \right )\right )\right ) \]

Problem 2597

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

Problem 2598

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

Problem 2599

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

Problem 2600

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