2.17.128 Problems 12701 to 12800

Problem 12701

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

Problem 12702

ODE	\[ \boxed {y^{\prime }-\frac {y}{-x^{2}+1}=\sqrt {x}} \] With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = 1\right ] \end {align*}
program solution	\[ y = \frac {\sqrt {3}\, x -3 x \left (\int _{\frac {1}{2}}^{x}\frac {\textit {\_a}^{\frac {5}{2}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )+3 x \left (\int _{\frac {1}{2}}^{x}\frac {\sqrt {\textit {\_a}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )+\sqrt {3}-3 \left (\int _{\frac {1}{2}}^{x}\frac {\textit {\_a}^{\frac {5}{2}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )+3 \left (\int _{\frac {1}{2}}^{x}\frac {\sqrt {\textit {\_a}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )}{3 \sqrt {-x^{2}+1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (12 i \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {6}}{2}, \frac {\sqrt {2}}{2}\right )-\sqrt {3}\, \sqrt {2}-8 i \operatorname {EllipticF}\left (\frac {\sqrt {3}}{2}, \sqrt {2}\right )+2 \sqrt {3}\right ) \left (1+x \right )}{6 \sqrt {-x^{2}+1}}+\frac {-2 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+2 x^{3}-2 x}{\sqrt {x}\, \left (3 x -3\right )} \]

Problem 12703

ODE	\[ \boxed {y^{\prime }-\frac {y}{-x^{2}+1}=\sqrt {x}} \]
program solution	\[ \int _{}^{x}-\frac {-\textit {\_a}^{\frac {5}{2}}+\sqrt {\textit {\_a}}+y}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (1+x \right ) c_{1}}{\sqrt {-x^{2}+1}}+\frac {-2 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+2 x^{3}-2 x}{\sqrt {x}\, \left (3 x -3\right )} \]

Problem 12704

ODE	\[ \boxed {y^{\prime }-\frac {y}{-x^{2}+1}=\sqrt {x}} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
program solution	\[ y = \frac {i x \sqrt {3}+i \sqrt {3}-3 x \left (\int _{2}^{x}\frac {\textit {\_a}^{\frac {5}{2}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )+3 x \left (\int _{2}^{x}\frac {\sqrt {\textit {\_a}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )-3 \left (\int _{2}^{x}\frac {\textit {\_a}^{\frac {5}{2}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )+3 \left (\int _{2}^{x}\frac {\sqrt {\textit {\_a}}}{\left (1+\textit {\_a} \right ) \sqrt {-\textit {\_a}^{2}+1}}d \textit {\_a} \right )}{3 \sqrt {-x^{2}+1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {i \left (1+x \right ) \left (-1+\frac {2 \left (\sqrt {3}\, \operatorname {EllipticF}\left (\sqrt {3}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {3}\, \operatorname {EllipticE}\left (\sqrt {3}, \frac {\sqrt {2}}{2}\right )+3\right ) \sqrt {2}}{3}\right ) \sqrt {3}}{3 \sqrt {-x^{2}+1}}+\frac {-2 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+6 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+2 x^{3}-2 x}{\sqrt {x}\, \left (3 x -3\right )} \]

Problem 12705

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

Problem 12706

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

Problem 12707

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

Problem 12708

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

Problem 12709

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

Problem 12710

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

Problem 12711

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

Problem 12712

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

Problem 12713

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

Problem 12714

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

Problem 12715

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

Problem 12716

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

Problem 12717

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

Problem 12718

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

Problem 12719

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

Problem 12720

ODE	\[ \boxed {y^{\prime }-3 x y^{\frac {1}{3}}=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -x^{3} \]

Problem 12721

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

Problem 12722

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

Problem 12723

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

Problem 12724

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

Problem 12725

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

Problem 12726

ODE	\[ \boxed {y^{\prime }-\sqrt {\left (y+2\right ) \left (y-1\right )}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -3] \end {align*}
program solution	\[ -\ln \left (2\right )+\ln \left (1+2 y+2 \sqrt {\left (y+2\right ) \left (y-1\right )}\right ) = x -\ln \left (2\right )+i \pi \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{x}}{4}-\frac {1}{2}-\frac {9 \,{\mathrm e}^{-x}}{4} \]

Problem 12727

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

Problem 12728

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

Problem 12729

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

Problem 12730

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

Problem 12731

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

Problem 12732

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

Problem 12733

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

Problem 12734

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

Problem 12735

ODE	\[ \boxed {y^{\prime }-x \sqrt {1-y^{2}}=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {9}{10}}\right ] \end {align*}
program solution	\[ y = \sin \left (\frac {x^{2}}{2}+\arcsin \left (\frac {9}{10}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (\frac {x^{2}}{2}+\arcsin \left (\frac {9}{10}\right )\right ) \]

Problem 12736

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

Problem 12737

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

Problem 12738

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

Problem 12739

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

Problem 12740

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

Problem 12741

ODE	\[ \boxed {y^{\prime }-\frac {\sqrt {x^{2}+4 y}}{2}=-\frac {x}{2}} \] With initial conditions \begin {align*} \left [y \left (1\right ) = -{\frac {1}{5}}\right ] \end {align*}
program solution	\[ y = \frac {3}{10}-\frac {\sqrt {5}}{10}-\frac {x}{2}+\frac {\sqrt {5}\, x}{10} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-1+x \right ) \sqrt {5}}{10}-\frac {x}{2}+\frac {3}{10} \\ y \left (x \right ) &= \frac {\left (\sqrt {5}-5\right ) \left (-5+\sqrt {5}+10 x \right )}{100} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (50+20 \sqrt {5}\right )^{\frac {1}{3}} \left (2^{\frac {1}{3}} x -\frac {\left (50+20 \sqrt {5}\right )^{\frac {1}{3}}}{5}\right )}{10} \\ \end{align*}

Problem 12742

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

Problem 12743

ODE	\[ \boxed {3 y^{\prime \prime }-2 y^{\prime }+4 y=x} \] With initial conditions \begin {align*} [y \left (-1\right ) = 2, y^{\prime }\left (-1\right ) = 3] \end {align*}
program solution	\[ y = \frac {\left (\left (49 \sqrt {11}\, \sin \left (\frac {\sqrt {11}}{3}\right )+187 \cos \left (\frac {\sqrt {11}}{3}\right )\right ) \cos \left (\frac {\sqrt {11}\, x}{3}\right )+49 \sin \left (\frac {\sqrt {11}\, x}{3}\right ) \left (\sqrt {11}\, \cos \left (\frac {\sqrt {11}}{3}\right )-\frac {187 \sin \left (\frac {\sqrt {11}}{3}\right )}{49}\right )\right ) {\mathrm e}^{\frac {x}{3}+\frac {1}{3}}}{88}+\frac {x}{4}+\frac {1}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (49 \sin \left (\frac {\sqrt {11}}{3}\right ) \sqrt {11}+187 \cos \left (\frac {\sqrt {11}}{3}\right )\right ) \cos \left (\frac {\sqrt {11}\, x}{3}\right )+49 \sin \left (\frac {\sqrt {11}\, x}{3}\right ) \left (\cos \left (\frac {\sqrt {11}}{3}\right ) \sqrt {11}-\frac {187 \sin \left (\frac {\sqrt {11}}{3}\right )}{49}\right )\right ) {\mathrm e}^{\frac {x}{3}+\frac {1}{3}}}{88}+\frac {x}{4}+\frac {1}{8} \]

Problem 12744

ODE	\[ \boxed {x y^{\prime \prime \prime }+y^{\prime } x=4} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 1, y^{\prime \prime }\left (1\right ) = -1] \end {align*}
program solution	\[ y = \left (4 \,\operatorname {Ci}\left (1\right )-4 \,\operatorname {Ci}\left (x \right )+\cos \left (1\right )-\sin \left (1\right )\right ) \cos \left (x \right )+\left (4 \,\operatorname {Si}\left (1\right )-4 \,\operatorname {Si}\left (x \right )+\cos \left (1\right )+\sin \left (1\right )\right ) \sin \left (x \right )+4 \ln \left (x \right )-1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (4 \,\operatorname {Ci}\left (1\right )-4 \,\operatorname {Ci}\left (x \right )+\cos \left (1\right )-\sin \left (1\right )\right ) \cos \left (x \right )+\left (4 \,\operatorname {Si}\left (1\right )-4 \,\operatorname {Si}\left (x \right )+\cos \left (1\right )+\sin \left (1\right )\right ) \sin \left (x \right )+4 \ln \left (x \right )-1 \]

Problem 12745

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

Problem 12746

ODE	\[ \boxed {x \left (x -3\right ) y^{\prime \prime }+3 y^{\prime }=x^{2}} \] With initial conditions \begin {align*} [y \left (5\right ) = 0, y^{\prime }\left (5\right ) = 1] \end {align*}
program solution	\[ y = \frac {x^{2}}{2}-\frac {8 x}{5}-\frac {24 \ln \left (x -3\right )}{5}-\frac {9}{2}+\frac {24 \ln \left (2\right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}}{2}-\frac {8 x}{5}-\frac {24 \ln \left (-3+x \right )}{5}+\frac {24 \ln \left (2\right )}{5}-\frac {9}{2} \]

Problem 12747

ODE	\[ \boxed {\sqrt {1-x}\, y^{\prime \prime }-4 y=\sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (-2\right ) = 3, y^{\prime }\left (-2\right ) = -1] \end {align*}
program solution	\[ y = 2 \left (\frac {\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3}-\frac {\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3}+\frac {\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}-\frac {\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}+\left (\left (-1+i\right ) 2^{\frac {3}{4}} \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {i \sqrt {2}\, \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}\right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )-\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\left (-1+i\right ) 2^{\frac {3}{4}} \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {i \sqrt {2}\, \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}\right )\right ) \sqrt {x}\, \pi \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 \left (\left (\left (1-x \right )^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (\left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}}}{3}\right ) \sqrt {-\textit {\_z1} +1}\, \sin \left (\textit {\_z1} \right )}{\left (\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}\right )^{\frac {1}{3}}}d \textit {\_z1} \right ) \sqrt {3}+6 \,3^{\frac {3}{4}} \operatorname {BesselI}\left (-\frac {1}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )-3 \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )\right ) \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{\frac {3}{2}}}}{3}\right )+\left (-1+x \right ) \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{\frac {3}{2}}}}{3}\right ) \left (6 \,3^{\frac {3}{4}} \operatorname {BesselI}\left (\frac {1}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )+\left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}}}{3}\right ) \left (\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}\right )^{\frac {1}{3}} \sin \left (\textit {\_z1} \right )}{\sqrt {-\textit {\_z1} +1}}d \textit {\_z1} \right ) \sqrt {3}-3 \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )\right )\right ) \pi }{9 \left (\left (1-x \right )^{\frac {3}{2}}\right )^{\frac {1}{3}}} \]

Problem 12748

ODE	\[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+\ln \left (x \right ) y=x \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 2] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12749

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 12750

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 12751

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

Problem 12752

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

Problem 12753

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

Problem 12754

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

Problem 12755

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

Problem 12756

ODE	\[ \boxed {y^{\prime \prime }-4 y=31} \] With initial conditions \begin {align*} [y \left (0\right ) = -9, y^{\prime }\left (0\right ) = 6] \end {align*}
program solution	\[ y = -\frac {31}{4}+\frac {7 \,{\mathrm e}^{2 x}}{8}-\frac {17 \,{\mathrm e}^{-2 x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {7 \,{\mathrm e}^{2 x}}{8}-\frac {17 \,{\mathrm e}^{-2 x}}{8}-\frac {31}{4} \]

Problem 12757

ODE	\[ \boxed {y^{\prime \prime }+9 y=27 x +18} \] With initial conditions \begin {align*} [y \left (0\right ) = 23, y^{\prime }\left (0\right ) = 21] \end {align*}
program solution	\[ y = 2+21 \cos \left (3 x \right )+6 \sin \left (3 x \right )+3 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = 6 \sin \left (3 x \right )+21 \cos \left (3 x \right )+3 x +2 \]

Problem 12758

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

Problem 12759

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

Problem 12760

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

Problem 12761

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

Problem 12762

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

Problem 12763

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

Problem 12764

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

Problem 12765

ODE	\[ \boxed {36 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }-11 y^{\prime \prime }+2 y^{\prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{-\frac {x}{3}} c_{1} +x \,{\mathrm e}^{-\frac {x}{3}} c_{2} +{\mathrm e}^{\frac {x}{2}} c_{3} +x \,{\mathrm e}^{\frac {x}{2}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{4} x +c_{3} \right ) {\mathrm e}^{\frac {5 x}{6}}+c_{2} x +c_{1} \right ) {\mathrm e}^{-\frac {x}{3}} \]

Problem 12766

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

Problem 12767

ODE	\[ \boxed {y^{\left (5\right )}-y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+35 y^{\prime \prime }+16 y^{\prime }-52 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+x \,{\mathrm e}^{-2 x} c_{2} +c_{3} {\mathrm e}^{x}+{\mathrm e}^{\left (2-3 i\right ) x} c_{4} +{\mathrm e}^{\left (2+3 i\right ) x} c_{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} {\mathrm e}^{4 x} \sin \left (3 x \right )+c_{5} {\mathrm e}^{4 x} \cos \left (3 x \right )+c_{1} {\mathrm e}^{3 x}+c_{3} x +c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 12768

ODE	\[ \boxed {y^{\left (8\right )}+8 y^{\prime \prime \prime \prime }+16 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (1+i\right ) x} c_{1} +x \,{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (-1+i\right ) x} c_{3} +x \,{\mathrm e}^{\left (-1+i\right ) x} c_{4} +{\mathrm e}^{\left (-1-i\right ) x} c_{5} +x \,{\mathrm e}^{\left (-1-i\right ) x} c_{6} +{\mathrm e}^{\left (1-i\right ) x} c_{7} +x \,{\mathrm e}^{\left (1-i\right ) x} c_{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{4} x +c_{2} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{3} x +c_{1} \right )\right ) {\mathrm e}^{-x}+{\mathrm e}^{x} \left (\left (c_{8} x +c_{6} \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{7} x +c_{5} \right )\right ) \]

Problem 12769

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

Problem 12770

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

Problem 12771

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

Problem 12772

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

Problem 12773

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-12 y^{\prime }+4 y=2 \,{\mathrm e}^{x}-4 \,{\mathrm e}^{2 x}} \]
program solution	\[ y = \left (c_{4} x +c_{3} \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \left (c_{2} x +c_{1} \right )+{\mathrm e}^{x} x^{2}-2 x^{2} {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-2 x^{2}+\left (c_{4} +8\right ) x +c_{2} -12\right ) {\mathrm e}^{2 x}+\left (x^{2}+\left (c_{3} +4\right ) x +c_{1} +6\right ) {\mathrm e}^{x} \]

Problem 12774

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime }=24 x^{2}-6 x +14+32 \cos \left (2 x \right )} \]
program solution	\[ y = c_{2} x +c_{1} +{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} +\frac {x^{2}}{4}+\frac {x^{4}}{2}+\frac {3 x}{8}-\frac {x^{3}}{4}-\frac {1}{8}-\frac {5 \cos \left (2 x \right )}{2}-2 x \sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-c_{1} -10\right ) \cos \left (2 x \right )}{4}+\frac {\left (-8 x -c_{2} \right ) \sin \left (2 x \right )}{4}+\frac {x^{4}}{2}-\frac {x^{3}}{4}+\frac {x^{2}}{4}+c_{3} x +c_{4} \]

Problem 12775

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

Problem 12776

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+3 y^{\prime \prime }-y^{\prime }=6 x -20-120 \,{\mathrm e}^{x} x^{2}} \]
program solution	\[ y = \left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{x}+c_{1} -3 x^{2}+2 x -40 x^{3} {\mathrm e}^{x}+10 x^{4} {\mathrm e}^{x}-2 x^{5} {\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-2 x^{5}+10 x^{4}-40 x^{3}+\left (c_{3} +120\right ) x^{2}+\left (c_{2} -2 c_{3} -240\right ) x +c_{1} -c_{2} +2 c_{3} +240\right ) {\mathrm e}^{x}-3 x^{2}+2 x +c_{4} \]

Problem 12777

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+21 y^{\prime }-26 y=36 \,{\mathrm e}^{2 x} \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+{\mathrm e}^{\left (2-3 i\right ) x} c_{2} +{\mathrm e}^{\left (2+3 i\right ) x} c_{3} -\frac {2 \,{\mathrm e}^{2 x} \left (3 \sin \left (3 x \right ) x +2 \cos \left (3 x \right )\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x} \left (3 c_{3} \sin \left (3 x \right )-6 x \sin \left (3 x \right )+3 c_{2} \cos \left (3 x \right )-2 \cos \left (3 x \right )+3 c_{1} \right )}{3} \]

Problem 12778

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

Problem 12779

ODE	\[ \boxed {y^{\left (6\right )}-12 y^{\left (5\right )}+63 y^{\prime \prime \prime \prime }-18 y^{\prime \prime \prime }+315 y^{\prime \prime }-300 y^{\prime }+125 y={\mathrm e}^{x} \left (48 \cos \left (x \right )+96 \sin \left (x \right )\right )} \]
program solution	\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =2\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =5\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =6\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =3\right ) x} c_{4} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =1\right ) x} c_{5} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{6}-12 \textit {\_Z}^{5}+63 \textit {\_Z}^{4}-18 \textit {\_Z}^{3}+315 \textit {\_Z}^{2}-300 \textit {\_Z} +125, \operatorname {index} =4\right ) x} c_{6} -\frac {48528 \,{\mathrm e}^{x} \cos \left (x \right )}{229205}-\frac {16896 \,{\mathrm e}^{x} \sin \left (x \right )}{229205} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 12780

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+12 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 5, y^{\prime \prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = -{\mathrm e}^{-2 x}+3 \,{\mathrm e}^{2 x}-{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-{\mathrm e}^{5 x}+3 \,{\mathrm e}^{4 x}-1\right ) {\mathrm e}^{-2 x} \]

Problem 12781

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

Problem 12782

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

Problem 12783

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

Problem 12784

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

Problem 12785

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

Problem 12786

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

Problem 12787

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

Problem 12788

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

Problem 12789

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=x \,{\mathrm e}^{x}-3 x^{2}} \]
program solution	\[ y = \frac {3 \left (-\frac {31}{162}+\frac {\left (x^{2}-\frac {2}{3} x +4 c_{1} +2 c_{2} -\frac {106}{9}\right ) {\mathrm e}^{3 x}}{9}+\left (x^{2}+x +\frac {3}{2}\right ) {\mathrm e}^{2 x}+\frac {2 c_{1}}{9}-\frac {2 c_{2}}{9}\right ) {\mathrm e}^{-2 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {9}{4}+\frac {3 x}{2}+\frac {3 x^{2}}{2}+\frac {{\mathrm e}^{x} \left (9 x^{2}+18 D\left (y \right )\left (0\right )+36 y \left (0\right )-6 x -106\right )}{54}+\frac {\left (36 y \left (0\right )-36 D\left (y \right )\left (0\right )-31\right ) {\mathrm e}^{-2 x}}{108} \]

Problem 12790

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+y^{\prime \prime }=x \,{\mathrm e}^{x}-3 x^{2}} \]
program solution	\[ y = -26-3 c_{3} +2 c_{4} +c_{1} -9 x^{2}-\frac {x^{4}}{4}-2 x^{3}+\frac {{\mathrm e}^{x} \left (x^{3}-6 c_{3} x +6 c_{4} x -6 x^{2}+18 c_{3} -12 c_{4} -18 x +156\right )}{6}+\left (-23+c_{2} -2 c_{3} +c_{4} \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = -26-\frac {x^{4}}{4}-9 x^{2}-2 x^{3}+y \left (0\right )+\frac {{\mathrm e}^{x} \left (x^{3}+6 x D^{\left (3\right )}\left (y \right )\left (0\right )-6 x D^{\left (2\right )}\left (y \right )\left (0\right )-6 x^{2}-12 D^{\left (3\right )}\left (y \right )\left (0\right )+18 D^{\left (2\right )}\left (y \right )\left (0\right )-18 x +156\right )}{6}-D^{\left (2\right )}\left (y \right )\left (0\right ) \left (3+2 x \right )+D^{\left (3\right )}\left (y \right )\left (0\right ) \left (x +2\right )+x \left (-23+D\left (y \right )\left (0\right )\right ) \]

Problem 12791

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

Problem 12792

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

Problem 12793

ODE	\[ \boxed {y^{\prime \prime }-9 y=x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\frac {2}{9}-\frac {x}{9}-\frac {7 \cosh \left (3 x \right )}{9}+\frac {10 \sinh \left (3 x \right )}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{9}-\frac {7 \cosh \left (3 x \right )}{9}+\frac {10 \sinh \left (3 x \right )}{27}-\frac {2}{9} \]

Problem 12794

ODE	\[ \boxed {y^{\prime \prime }+9 y=x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {2}{9}-\frac {11 \cos \left (3 x \right )}{9}+\frac {8 \sin \left (3 x \right )}{27}+\frac {x}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{9}-\frac {11 \cos \left (3 x \right )}{9}+\frac {8 \sin \left (3 x \right )}{27}+\frac {2}{9} \]

Problem 12795

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }+6 y=-2 \sin \left (3 x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = -\frac {13 \sqrt {23}\, {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {13 \,{\mathrm e}^{\frac {x}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, x}{2}\right )}{69}+\frac {{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {23}\, x}{2}\right )}{3}+\frac {\sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right )}{3} \]

Problem 12796

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

Problem 12797

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }=x +\cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {11}{8}+\frac {x^{2}}{4}-\frac {{\mathrm e}^{-x}}{2}+\frac {17 \,{\mathrm e}^{-2 x}}{40}-\frac {3 x}{4}-\frac {3 \cos \left (x \right )}{10}+\frac {\sin \left (x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 \cos \left (x \right )}{10}+\frac {\sin \left (x \right )}{10}-\frac {{\mathrm e}^{-x}}{2}-\frac {3 x}{4}+\frac {x^{2}}{4}+\frac {17 \,{\mathrm e}^{-2 x}}{40}+\frac {11}{8} \]

Problem 12798

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

Problem 12799

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

Problem 12800

ODE	\[ \boxed {y^{\prime \prime }+9 y=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {1}{9}-\frac {\cos \left (3 x \right )}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\cos \left (3 x \right )}{9}+\frac {1}{9} \]