Problems 5901 to 6000

Problem 5901


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )+2 x_{2} \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{3 t} \left (2 c_{1} \cos \left (2 t \right )-c_{2} \cos \left (2 t \right )-c_{1} \sin \left (2 t \right )-2 c_{2} \sin \left (2 t \right )\right ) \\ \end{align}

Problem 5902


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end {align}

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{-3 t} \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-t}-c_{1} {\mathrm e}^{-3 t} \\ \end{align}

Problem 5903


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )+2 \,{\mathrm e}^{-t}\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+3 t \end {align}

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= -c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} +\frac {{\mathrm e}^{-t}}{2}-\frac {4}{3}+t \,{\mathrm e}^{-t}+t \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} -\frac {{\mathrm e}^{-t}}{2}-\frac {5}{3}+2 t +t \,{\mathrm e}^{-t} \\ \end{align}

Problem 5904


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=16 x_{1} \left (t \right )-5 x_{2} \left (t \right ) \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (3 t +1\right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (12 t +1\right ) \\ \end{align}

Problem 5905


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right ) \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 0] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= -2 \,{\mathrm e}^{-2 t}+3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -3 \,{\mathrm e}^{-2 t}+3 \,{\mathrm e}^{-t} \\ \end{align}

Problem 5906


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-18 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-9 x_{2} \left (t \right ) \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 2] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (-30 t +1\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (-180 t +36\right )}{18} \\ \end{align}

Problem 5907


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+3 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 2] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (3 t +1\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (9 t +6\right )}{3} \\ \end{align}

Problem 5908


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-18 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-9 x_{2} \left (t \right ) \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 2, x_{2} \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (-6 t +2\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (-36 t +18\right )}{18} \\ \end{align}

Problem 5909


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 1] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{2 t} \\ \end{align}

Problem 5910


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )-8\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+3 \end {align}

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{2 t}}{2}-\frac {11 t}{2}+c_{2} \\ x_{2} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{2 t}}{2}+\frac {5}{2}+\frac {11 t}{2}-c_{2} \\ \end{align}

Problem 5911


ODE	\begin {align} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )-8\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+3 \end {align} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 2] \]

program solution

Maple solution	\begin{align} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{4}-\frac {11 t}{2}+\frac {3}{4} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{4}+\frac {7}{4}+\frac {11 t}{2} \\ \end{align}

Problem 5912


ODE	\[ \boxed {y^{\prime }={\mathrm e}^{3 x}+\sin \left (x \right )} \]

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

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{3 x}}{3}-\cos \left (x \right )+c_{1} \]

Problem 5913


ODE	\[ \boxed {y^{\prime \prime }=x +2} \]

program solution	\[ y = \frac {1}{6} x^{3}+x^{2}+c_{1} x +c_{2} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {1}{6} x^{3}+x^{2}+c_{1} x +c_{2} \]

Problem 5914


ODE	\[ \boxed {y^{\prime \prime \prime }=x^{2}} \]

program solution	\[ y = c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{60} x^{5} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {1}{60} x^{5}+\frac {1}{2} c_{1} x^{2}+c_{2} x +c_{3} \]

Problem 5915


ODE	\[ \boxed {y^{\prime }+y \cos \left (x \right )=0} \]

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

Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\sin \left (x \right )} \]

Problem 5916


ODE	\[ \boxed {y^{\prime }+y \cos \left (x \right )=\cos \left (x \right ) \sin \left (x \right )} \]

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

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

Problem 5917


ODE	\[ \boxed {y^{\prime \prime }-y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{x}}{2} \] Veriﬁed OK.

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

Problem 5918


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

Problem 5919


ODE	\[ \boxed {y^{\prime \prime }+k^{2} y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{\sqrt {-k^{2}}\, x}+\frac {c_{2} \sqrt {-k^{2}}\, {\mathrm e}^{-\sqrt {-k^{2}}\, x}}{2 k^{2}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (k x \right )+c_{2} \cos \left (k x \right ) \]

Problem 5920


ODE	\[ \boxed {y^{\prime }+5 y=2} \]

program solution	\[ y = \frac {{\mathrm e}^{-5 x}}{c_{1}^{5}}+\frac {2}{5} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {2}{5}+{\mathrm e}^{-5 x} c_{1} \]

Problem 5921


ODE	\[ \boxed {y^{\prime \prime }=3 x +1} \]

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

Maple solution	\[ y \left (x \right ) = \frac {1}{2} x^{3}+\frac {1}{2} x^{2}+c_{1} x +c_{2} \]

Problem 5922


ODE	\[ \boxed {y^{\prime }-y k=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{x k} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{k x} \]

Problem 5923


ODE	\[ \boxed {-2 y+y^{\prime }=1} \]

program solution	\[ y = {\mathrm e}^{2 x} c_{1}^{2}-\frac {1}{2} \] Veriﬁed OK.

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

Problem 5924


ODE	\[ \boxed {y+y^{\prime }={\mathrm e}^{x}} \]

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

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

Problem 5925


ODE	\[ \boxed {-2 y+y^{\prime }=x^{2}+x} \]

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

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

Problem 5926


ODE	\[ \boxed {y+3 y^{\prime }=2 \,{\mathrm e}^{-x}} \]

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

Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-x}+{\mathrm e}^{-\frac {x}{3}} c_{1} \]

Problem 5927


ODE	\[ \boxed {y^{\prime }+3 y={\mathrm e}^{i x}} \]

program solution	\[ \int _{}^{x}\left (3 y-{\mathrm e}^{i \textit {\_a}}\right ) {\mathrm e}^{3 \textit {\_a}}d \textit {\_a} = c_{1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-3 x} \left (\left (-3+i\right ) {\mathrm e}^{\left (3+i\right ) x}-10 c_{1} \right )}{10} \]

Problem 5928


ODE	\[ \boxed {y^{\prime }+i y=x} \]

program solution	\[ \int _{}^{x}\left (i y-\textit {\_a} \right ) {\mathrm e}^{i \textit {\_a}}d \textit {\_a} = c_{1} \] Veriﬁed OK.

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

Problem 5929


ODE	\[ \boxed {L y^{\prime }+R y=E} \]

program solution	\[ y = \frac {-\frac {{\mathrm e}^{-\frac {R x}{L}}}{c_{1}}+E}{R} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {R x}{L}} c_{1} R +E}{R} \]

Problem 5930


ODE	\[ \boxed {L y^{\prime }+R y=E \sin \left (\omega x \right )} \] With initial conditions \begin {align} [y \left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {-E L \,{\mathrm e}^{-\frac {R x}{L}} \cos \left (\omega x \right ) {\mathrm e}^{\frac {R x}{L}} \omega +E \,{\mathrm e}^{-\frac {R x}{L}} \sin \left (\omega x \right ) {\mathrm e}^{\frac {R x}{L}} R +E L \,{\mathrm e}^{-\frac {R x}{L}} \omega }{\omega ^{2} L^{2}+R^{2}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {E \left ({\mathrm e}^{-\frac {R x}{L}} L \omega -L \cos \left (\omega x \right ) \omega +\sin \left (\omega x \right ) R \right )}{\omega ^{2} L^{2}+R^{2}} \]

Problem 5931


ODE	\[ \boxed {L y^{\prime }+R y=E \,{\mathrm e}^{i \omega x}} \] With initial conditions \begin {align} [y \left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {E \left (\int _{0}^{x}{\mathrm e}^{\frac {\textit {\_a} \left (i L \omega +R \right )}{L}}d \textit {\_a} \right ) {\mathrm e}^{-\frac {R x}{L}}}{L} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {E \left ({\mathrm e}^{\frac {x \left (i L \omega +R \right )}{L}}-1\right ) {\mathrm e}^{-\frac {R x}{L}}}{i L \omega +R} \]

Problem 5932


ODE	\[ \boxed {y^{\prime }+a y=b \left (x \right )} \]

program solution	\[ \int _{}^{x}\left (a y-b \left (\textit {\_a} \right )\right ) {\mathrm e}^{a \textit {\_a}}d \textit {\_a} = c_{1} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left (\int b \left (x \right ) {\mathrm e}^{a x}d x +c_{1} \right ) {\mathrm e}^{-a x} \]

Problem 5933


ODE	\[ \boxed {2 y x +y^{\prime }=x} \]

program solution	\[ y = \frac {{\mathrm e}^{-x^{2}-2 c_{1}}}{2}+\frac {1}{2} \] Veriﬁed OK.

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

Problem 5934


ODE	\[ \boxed {y^{\prime } x +y=3 x^{3}-1} \]

program solution	\[ y = \frac {3 x^{4}+4 c_{1} -4 x}{4 x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\frac {3}{4} x^{4}-x +c_{1}}{x} \]

Problem 5935


ODE	\[ \boxed {y^{\prime }+{\mathrm e}^{x} y=3 \,{\mathrm e}^{x}} \]

program solution	\[ y = {\mathrm e}^{-{\mathrm e}^{x}-c_{1}}+3 \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 3+{\mathrm e}^{-{\mathrm e}^{x}} c_{1} \]

Problem 5936


ODE	\[ \boxed {y^{\prime }-y \tan \left (x \right )={\mathrm e}^{\sin \left (x \right )}} \]

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

Maple solution	\[ y \left (x \right ) = \sec \left (x \right ) \left ({\mathrm e}^{\sin \left (x \right )}+c_{1} \right ) \]

Problem 5937


ODE	\[ \boxed {2 y x +y^{\prime }={\mathrm e}^{-x^{2}} x} \]

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

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

Problem 5938


ODE	\[ \boxed {y^{\prime }+y \cos \left (x \right )={\mathrm e}^{-\sin \left (x \right )}} \] With initial conditions \begin {align} [y \left (\pi \right ) = \pi ] \end {align}

program solution	\[ y = {\mathrm e}^{-\sin \left (x \right )} x \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\sin \left (x \right )} x \]

Problem 5939


ODE	\[ \boxed {x^{2} y^{\prime }+2 y x=1} \]

program solution	\[ y = \frac {x +c_{1}}{x^{2}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {x +c_{1}}{x^{2}} \]

Problem 5940


ODE	\[ \boxed {y^{\prime }+2 y=b \left (x \right )} \]

program solution	\[ \int _{}^{x}\left (2 y-b \left (\textit {\_a} \right )\right ) {\mathrm e}^{2 \textit {\_a}}d \textit {\_a} = c_{1} \] Veriﬁed OK.

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

Problem 5941


ODE	\[ \boxed {y^{\prime }-y=1} \] With initial conditions \begin {align} [y \left (0\right ) = 0] \end {align}

program solution	\[ \ln \left (y+1\right ) = x \] Veriﬁed OK.

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

Problem 5942


ODE	\[ \boxed {y^{\prime }-y^{2}=1} \] With initial conditions \begin {align} [y \left (0\right ) = 0] \end {align}

program solution	\[ \arctan \left (y\right ) = x \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \tan \left (x \right ) \]

Problem 5943


ODE	\[ \boxed {y^{\prime }-y^{2}=1} \] With initial conditions \begin {align} [y \left (0\right ) = 0] \end {align}

program solution	\[ \arctan \left (y\right ) = x \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \tan \left (x \right ) \]

Problem 5944


ODE	\[ \boxed {y^{\prime \prime }-4 y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4} \] Veriﬁed OK.

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

Problem 5945


ODE	\[ \boxed {3 y^{\prime \prime }+2 y=0} \]

program solution	\[ y = c_{1} \cos \left (\frac {\sqrt {6}\, x}{3}\right )+\frac {c_{2} \sqrt {6}\, \sin \left (\frac {\sqrt {6}\, x}{3}\right )}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\frac {\sqrt {6}\, x}{3}\right )+c_{2} \cos \left (\frac {\sqrt {6}\, x}{3}\right ) \]

Problem 5946


ODE	\[ \boxed {y^{\prime \prime }+16 y=0} \]

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

Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (4 x \right )+c_{2} \cos \left (4 x \right ) \]

Problem 5947


ODE	\[ \boxed {y^{\prime \prime }=0} \]

program solution	\[ y = c_{1} x +c_{2} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} x +c_{2} \]

Problem 5948


ODE	\[ \boxed {y^{\prime \prime }+2 i y^{\prime }+y=0} \]

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

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

Problem 5949


ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+5 y=0} \]

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

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

Problem 5950


ODE	\[ \boxed {y^{\prime \prime }+\left (-1+3 i\right ) y^{\prime }-3 i y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{x}+\left (-\frac {1}{10}+\frac {3 i}{10}\right ) c_{2} {\mathrm e}^{-3 i x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{-3 i x} \]

Problem 5951


ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {3 \,{\mathrm e}^{2 x}}{5}+\frac {2 \,{\mathrm e}^{-3 x}}{5} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (3 \,{\mathrm e}^{5 x}+2\right ) {\mathrm e}^{-3 x}}{5} \]

Problem 5952


ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = \frac {\left ({\mathrm e}^{5 x}-1\right ) {\mathrm e}^{-3 x}}{5} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{5 x}-1\right ) {\mathrm e}^{-3 x}}{5} \]

Problem 5953


ODE	\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 1, y \left (\frac {\pi }{2}\right ) = 2\right ] \end {align}

program solution	\[ y = \cos \left (x \right )+2 \sin \left (x \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 2 \sin \left (x \right )+\cos \left (x \right ) \]

Problem 5954


ODE	\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y \left (\pi \right ) = 0] \end {align}

program solution	\[ y = c_{2} \sin \left (x \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (x \right ) \]

Problem 5955


ODE	\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 0, y^{\prime }\left (\frac {\pi }{2}\right ) = 0\right ] \end {align}

program solution	\[ y = c_{2} \sin \left (x \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (x \right ) \]

Problem 5956


ODE	\[ \boxed {y^{\prime \prime }+y=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 0, y \left (\frac {\pi }{2}\right ) = 0\right ] \end {align}

program solution	\[ y = 0 \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 0 \]

Problem 5957


ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = \frac {{\mathrm e}^{3 x}}{4}-\frac {{\mathrm e}^{-x}}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{3 x}}{4}-\frac {{\mathrm e}^{-x}}{4} \]

Problem 5958


ODE	\[ \boxed {y^{\prime \prime }+\left (1+4 i\right ) y^{\prime }+y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = 0 \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 0 \]

Problem 5959


ODE	\[ \boxed {y^{\prime \prime }+\left (-1+3 i\right ) y^{\prime }-3 i y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \left (\frac {1}{5}-\frac {3 i}{5}\right ) {\mathrm e}^{-3 i x}+\left (\frac {9}{5}+\frac {3 i}{5}\right ) {\mathrm e}^{x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left (\frac {9}{5}+\frac {3 i}{5}\right ) {\mathrm e}^{x}+\left (\frac {1}{5}-\frac {3 i}{5}\right ) {\mathrm e}^{-3 i x} \]

Problem 5960


ODE	\[ \boxed {y^{\prime \prime }+10 y=0} \] With initial conditions \begin {align} [y \left (0\right ) = \pi , y^{\prime }\left (0\right ) = \pi ^{2}] \end {align}

program solution	\[ y = \cos \left (\sqrt {10}\, x \right ) \pi +\frac {\sin \left (\sqrt {10}\, x \right ) \sqrt {10}\, \pi ^{2}}{10} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\pi \left (\pi \sqrt {10}\, \sin \left (\sqrt {10}\, x \right )+10 \cos \left (\sqrt {10}\, x \right )\right )}{10} \]

Problem 5961


ODE	\[ \boxed {y^{\prime \prime }+4 y=\cos \left (x \right )} \]

program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {\cos \left (x \right )}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{1} +\frac {\cos \left (x \right )}{3} \]

Problem 5962


ODE	\[ \boxed {y^{\prime \prime }+9 y=\sin \left (3 x \right )} \]

program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-\frac {x \cos \left (3 x \right )}{6} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\left (-x +6 c_{1} \right ) \cos \left (3 x \right )}{6}+\sin \left (3 x \right ) c_{2} \]

Problem 5963


ODE	\[ \boxed {y^{\prime \prime }+y=\tan \left (x \right )} \]

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

Maple solution	\[ y \left (x \right ) = \sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} -\cos \left (x \right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right ) \]

Problem 5964


ODE	\[ \boxed {y^{\prime \prime }+2 i y^{\prime }+y=x} \]

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

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

Problem 5965


ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+5 y=3 \,{\mathrm e}^{-x}+2 x^{2}} \]

program solution	\[ y = {\mathrm e}^{2 x} \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+\frac {3 \,{\mathrm e}^{-x}}{10}+\frac {44}{125}+\frac {16 x}{25}+\frac {2 x^{2}}{5} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \sin \left (x \right ) c_{2} +{\mathrm e}^{2 x} \cos \left (x \right ) c_{1} +\frac {3 \,{\mathrm e}^{-x}}{10}+\frac {2 x^{2}}{5}+\frac {16 x}{25}+\frac {44}{125} \]

Problem 5966


ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+6 y=\sin \left (x \right )} \]

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

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{6 x} c_{2} +{\mathrm e}^{x} c_{1} +\frac {7 \cos \left (x \right )}{74}+\frac {5 \sin \left (x \right )}{74} \]

Problem 5967


ODE	\[ \boxed {y^{\prime \prime }+y=2 \sin \left (2 x \right ) \sin \left (x \right )} \]

program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+\frac {x \sin \left (x \right )}{2}+\frac {\cos \left (3 x \right )}{8} \] Veriﬁed OK.

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

Problem 5968


ODE	\[ \boxed {y^{\prime \prime }+y=\sec \left (x \right )} \]

program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )+\ln \left (\cos \left (x \right )\right ) \cos \left (x \right )+x \sin \left (x \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\ln \left (\sec \left (x \right )\right ) \cos \left (x \right )+\cos \left (x \right ) c_{1} +\sin \left (x \right ) \left (x +c_{2} \right ) \]

Problem 5969


ODE	\[ \boxed {4 y^{\prime \prime }-y={\mathrm e}^{x}} \]

program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}}+c_{2} {\mathrm e}^{\frac {x}{2}}+\frac {{\mathrm e}^{x}}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{2}} c_{2} +c_{1} {\mathrm e}^{-\frac {x}{2}}+\frac {{\mathrm e}^{x}}{3} \]

Problem 5970


ODE	\[ \boxed {6 y^{\prime \prime }+5 y^{\prime }-6 y=x} \]

program solution	\[ y = c_{1} {\mathrm e}^{-\frac {3 x}{2}}+\frac {6 c_{2} {\mathrm e}^{\frac {2 x}{3}}}{13}-\frac {x}{6}-\frac {5}{36} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (x +\frac {5}{6}\right ) {\mathrm e}^{\frac {3 x}{2}}-6 \,{\mathrm e}^{\frac {13 x}{6}} c_{2} -6 c_{1} \right ) {\mathrm e}^{-\frac {3 x}{2}}}{6} \]

Problem 5971


ODE	\[ \boxed {y^{\prime \prime }+\omega ^{2} y=A \cos \left (\omega x \right )} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = \frac {2 \,{\mathrm e}^{-\sqrt {-\omega ^{2}}\, x} \sqrt {-\omega ^{2}}\, \omega -\omega \left (A +2 \sqrt {-\omega ^{2}}\right ) {\mathrm e}^{\sqrt {-\omega ^{2}}\, x}+A \sin \left (\omega x \right ) \sqrt {-\omega ^{2}}+2 \left (x \omega \sin \left (\omega x \right )+\frac {\cos \left (\omega x \right )}{2}\right ) \omega A}{4 \omega ^{3}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\sin \left (\omega x \right ) \left (1+\frac {A x}{2}\right )}{\omega } \]

Problem 5972


ODE	\[ \boxed {y^{\prime \prime \prime }-8 y=0} \]

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

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{-x} \sin \left (\sqrt {3}\, x \right )+c_{3} {\mathrm e}^{-x} \cos \left (\sqrt {3}\, x \right ) \]

Problem 5973


ODE	\[ \boxed {y^{\prime \prime \prime \prime }+16 y=0} \]

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

Maple solution	\[ y \left (x \right ) = -c_{1} {\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )-c_{2} {\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_{3} {\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+c_{4} {\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \]

Problem 5974


ODE	\[ \boxed {y^{\prime \prime \prime }-5 y^{\prime \prime }+6 y^{\prime }=0} \]

program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{3 x} \]

Problem 5975


ODE	\[ \boxed {y^{\prime \prime \prime }-i y^{\prime \prime }+4 y^{\prime }-4 i y=0} \]

program solution	\[ y = {\mathrm e}^{i x} c_{1} +{\mathrm e}^{2 i x} c_{2} +{\mathrm e}^{-2 i x} c_{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{i x}+c_{2} {\mathrm e}^{2 i x}+c_{3} {\mathrm e}^{-2 i x} \]

Problem 5976


ODE	\[ \boxed {y^{\prime \prime \prime \prime }+5 y^{\prime \prime }+4 y=0} \]

program solution	\[ y = {\mathrm e}^{i x} c_{1} +c_{2} {\mathrm e}^{-i x}+{\mathrm e}^{2 i x} c_{3} +{\mathrm e}^{-2 i x} c_{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 2 c_{2} \cos \left (x \right )^{2}+\left (2 c_{1} \sin \left (x \right )+c_{4} \right ) \cos \left (x \right )+c_{3} \sin \left (x \right )-c_{2} \]

Problem 5977


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	\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{-2 x}+c_{3} \sin \left (2 x \right )+c_{4} \cos \left (2 x \right ) \]

Problem 5978


ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime }-2 y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+{\mathrm e}^{2 x} c_{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-x}+{\mathrm e}^{2 x} c_{1} \]

Problem 5979


ODE	\[ \boxed {y^{\prime \prime \prime }-3 i y^{\prime \prime }-3 y^{\prime }+i y=0} \]

program solution	\[ y = {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +x^{2} {\mathrm e}^{i x} c_{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{i x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]

Problem 5980


ODE	\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime }=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -\frac {{\mathrm e}^{-2 x}}{4}+\frac {{\mathrm e}^{2 x}}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x}}{4}-\frac {{\mathrm e}^{-2 x}}{4} \]

Problem 5981


ODE	\[ \boxed {y^{\left (5\right )}-y^{\prime \prime \prime \prime }-y^{\prime }+y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime }\left (0\right ) = 0, y^{\prime \prime \prime \prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = \frac {{\mathrm e}^{-x}}{8}+\frac {5 \,{\mathrm e}^{x}}{8}-\frac {x \,{\mathrm e}^{x}}{4}+\frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right )}{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x}}{8}+\frac {\left (-2 x +5\right ) {\mathrm e}^{x}}{8}+\frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right )}{4} \]

Problem 5982


ODE	\[ \boxed {y^{\prime \prime }+y=0} \]

program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (x \right )+\cos \left (x \right ) c_{2} \]

Problem 5983


ODE	\[ \boxed {y^{\prime \prime }-y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{x}}{2} \] Veriﬁed OK.

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

Problem 5984


ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} +c_{3} \sin \left (x \right )+c_{4} \cos \left (x \right ) \]

Problem 5985


ODE	\[ \boxed {y^{\left (5\right )}+2 y=0} \]

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

Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\left (-i 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}+2^{\frac {1}{5}} \sqrt {5}+2^{\frac {1}{5}}\right ) x}{4}}+c_{2} {\mathrm e}^{-\frac {x \left (i \left (\sqrt {5}+1\right ) 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}+2 \,2^{\frac {1}{5}} \left (\sqrt {5}-1\right )\right )}{8}}+c_{3} {\mathrm e}^{-2^{\frac {1}{5}} x}+c_{4} {\mathrm e}^{\frac {\left (i \left (\sqrt {5}+1\right ) 2^{\frac {7}{10}} \sqrt {5-\sqrt {5}}-2 \,2^{\frac {1}{5}} \left (\sqrt {5}-1\right )\right ) x}{8}}+c_{5} {\mathrm e}^{2^{\frac {1}{5}} \left (\cos \left (\frac {\pi }{5}\right )+i \sin \left (\frac {\pi }{5}\right )\right ) x} \]

Problem 5986


ODE	\[ \boxed {y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+4 y=0} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{-2 x}+{\mathrm e}^{x} c_{3} +{\mathrm e}^{2 x} c_{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{4 x} c_{1} +c_{4} {\mathrm e}^{3 x}+{\mathrm e}^{x} c_{2} +c_{3} \right ) {\mathrm e}^{-2 x} \]

Problem 5987


ODE	\[ \boxed {y^{\prime \prime \prime }+y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 0] \end {align}

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

Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{\frac {3 x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}+{\mathrm e}^{\frac {3 x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )-1\right ) {\mathrm e}^{-x}}{3} \]

Problem 5988


ODE	\[ \boxed {y^{\prime \prime \prime }-i y^{\prime \prime }+y^{\prime }-i y=0} \]

program solution	\[ y = {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{i x}+{\mathrm e}^{-i x} c_{1} \]

Problem 5989


ODE	\[ \boxed {y^{\prime \prime }-2 i y^{\prime }-y=0} \]

program solution	\[ y = {\mathrm e}^{i x} c_{1} +x \,{\mathrm e}^{i x} c_{2} \] Veriﬁed OK.

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

Problem 5990


ODE	\[ \boxed {y^{\prime \prime \prime \prime }-k^{4} y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 0] \end {align}

program solution	\[ y = 0 \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 0 \]

Problem 5991


ODE	\[ \boxed {y^{\prime \prime \prime }-y=x} \]

program solution	\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{3} -x \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -x +{\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \]

Problem 5992


ODE	\[ \boxed {y^{\prime \prime \prime }-8 y={\mathrm e}^{i x}} \]

program solution	\[ y = c_{1} {\mathrm e}^{2 x}+{\mathrm e}^{\left (-i \sqrt {3}-1\right ) x} c_{2} +{\mathrm e}^{\left (i \sqrt {3}-1\right ) x} c_{3} +\left (-\frac {8}{65}+\frac {i}{65}\right ) \cos \left (x \right )+\left (-\frac {1}{65}-\frac {8 i}{65}\right ) \sin \left (x \right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \left (-\frac {8}{65}+\frac {i}{65}\right ) {\mathrm e}^{i x}+{\mathrm e}^{2 x} c_{1} +c_{2} {\mathrm e}^{-x} \cos \left (\sqrt {3}\, x \right )+c_{3} {\mathrm e}^{-x} \sin \left (\sqrt {3}\, x \right ) \]

Problem 5993


ODE	\[ \boxed {y^{\prime \prime \prime \prime }+16 y=\cos \left (x \right )} \]

program solution	\[ y = {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{4} +\frac {\cos \left (x \right )}{17} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{4} {\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_{2} {\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_{3} {\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+c_{1} {\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+\frac {\cos \left (x \right )}{17} \]

Problem 5994


ODE	\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+y={\mathrm e}^{x}} \]

program solution	\[ y = {\mathrm e}^{x} \left (c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \right )+\frac {{\mathrm e}^{x} x^{4}}{24} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (\frac {1}{24} x^{4}+c_{1} +c_{2} x +c_{3} x^{2}+x^{3} c_{4} \right ) \]

Problem 5995


ODE	\[ \boxed {y^{\prime \prime \prime \prime }-y=\cos \left (x \right )} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{3} +{\mathrm e}^{-i x} c_{4} -\frac {5 \cos \left (x \right )}{16}+\frac {\left (i-4 x \right ) \sin \left (x \right )}{16} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{4} {\mathrm e}^{-x}+\frac {\left (4 c_{1} -1\right ) \cos \left (x \right )}{4}+\frac {\left (-x +4 c_{3} \right ) \sin \left (x \right )}{4}+{\mathrm e}^{x} c_{2} \]

Problem 5996


ODE	\[ \boxed {y^{\prime \prime }-2 i y^{\prime }-y={\mathrm e}^{i x}-2 \,{\mathrm e}^{-i x}} \]

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

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

Problem 5997


ODE	\[ \boxed {y^{\prime \prime }+4 y=\cos \left (x \right )} \]

program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {\cos \left (x \right )}{3} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{1} +\frac {\cos \left (x \right )}{3} \]

Problem 5998


ODE	\[ \boxed {y^{\prime \prime }+4 y=\sin \left (2 x \right )} \]

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

Maple solution	\[ y \left (x \right ) = \frac {\left (-x +4 c_{1} \right ) \cos \left (2 x \right )}{4}+\sin \left (2 x \right ) c_{2} \]

Problem 5999


ODE	\[ \boxed {y^{\prime \prime }-4 y=3 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{-x}} \]

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

Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x} \left (\frac {\left (12 x +16 c_{2} -3\right ) {\mathrm e}^{4 x}}{16}+c_{1} -\frac {4 \,{\mathrm e}^{x}}{3}\right ) \]

Problem 6000


ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=x^{2}+\cos \left (x \right )} \]

program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{2 x}}{3}-\frac {3 \cos \left (x \right )}{10}-\frac {\sin \left (x \right )}{10}-\frac {3}{4}+\frac {x}{2}-\frac {x^{2}}{2} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{2 x}+c_{1} {\mathrm e}^{-x}-\frac {x^{2}}{2}-\frac {3 \cos \left (x \right )}{10}-\frac {\sin \left (x \right )}{10}+\frac {x}{2}-\frac {3}{4} \]

2.17.60 Problems 5901 to 6000