2.17.153 Problems 15201 to 15300

Problem 15201

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

Problem 15202

ODE	\[ \boxed {y^{\prime \prime }-\sqrt {y^{\prime }+1}=0} \]
program solution	\[ \int _{}^{y}\frac {4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+4 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}+16}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}+\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}+16-8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = c_{3} +x \] Verified OK. \[ \int _{}^{y}\frac {8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}}{i \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}} \sqrt {3}-\left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {4}{3}}-16+8 \left (6 \textit {\_a} +6 c_{1} +2 \sqrt {9 \textit {\_a}^{2}+18 \textit {\_a} c_{1} +9 c_{1}^{2}-16}\right )^{\frac {2}{3}}-16 i \sqrt {3}}d \textit {\_a} = x +c_{4} \] Verified OK.
Maple solution	\begin{align*} y &= -x +c_{1} \\ y &= \frac {1}{12} x^{3}+\frac {1}{4} c_{1} x^{2}+\frac {1}{4} c_{1}^{2} x -x +c_{2} \\ \end{align*}

Problem 15203

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } \ln \left (y^{\prime }\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = x \] Verified OK.
Maple solution	\[ y = -\operatorname {expIntegral}_{1}\left (-2 i \pi \_Z5 \,{\mathrm e}^{x}\right )+\operatorname {expIntegral}_{1}\left (-2 i \pi \_Z5 \right ) \]

Problem 15204

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

Problem 15205

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

Problem 15206

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

Problem 15207

ODE
program solution
Maple solution	\[ y = \ln \left (x +c_{1} \right ) \left (x +c_{1} \right )+\left (c_{2} -1\right ) x -c_{1} +c_{3} \]

Problem 15208

ODE	\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}=0} \]
program solution	\[ y = c_{2} {\mathrm e}^{c_{1} x} \] Verified OK.
Maple solution	\begin{align*} y &= 0 \\ y &= {\mathrm e}^{c_{1} x} c_{2} \\ \end{align*}

Problem 15209

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

Problem 15210

ODE	\[ \boxed {3 y^{\prime } y^{\prime \prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {1}{27} x^{3}+\frac {1}{3} x^{2}+x +1 \] Verified OK. \[ y = -\frac {i \left (2 i x^{3}-9 i x^{2}+9 \sqrt {3}\, x^{2}-27 i x -27 \sqrt {3}\, x +54 i\right )}{54} \] Warning, solution could not be verified \[ y = \frac {i x^{2} \sqrt {3}}{6}-\frac {i \sqrt {3}\, x}{2}+\frac {x^{3}}{27}-\frac {x^{2}}{6}-\frac {x}{2}+1 \] Warning, solution could not be verified
Maple solution	\[ y = \frac {\left (x +3\right )^{3}}{27} \]

Problem 15211

ODE	\[ \boxed {2 y^{\prime \prime }-3 y^{2}=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = 1, y^{\prime }\left (-2\right ) = -1] \end {align*}
program solution	\[ y = \frac {4}{x^{2}} \] Warning, solution could not be verified \[ y = \frac {4}{\left (4+x \right )^{2}} \] Verified OK.
Maple solution	\[ y = \frac {4}{\left (x +4\right )^{2}} \]

Problem 15212

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

Problem 15213

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

Problem 15214

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

Problem 15215

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

Problem 15216

ODE	\[ \boxed {y^{3} y^{\prime \prime }=-1} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = 1 \] Warning, solution could not be verified
Maple solution	\[ \text {No solution found} \]

Problem 15217

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

Problem 15218

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

Problem 15219

ODE	\[ \boxed {2 y y^{\prime \prime }-3 {y^{\prime }}^{2}-4 y^{2}=0} \]
program solution	\[ y = \frac {4 \left (\tan \left (x +c_{4} \right )^{2}+1\right ) {\mathrm e}^{-2 c_{2}}}{c_{3}^{2}} \] Verified OK. \[ y = \frac {4 \left (\tan \left (x +\textit {\_C5} \right )^{2}+1\right ) {\mathrm e}^{-2 c_{2}}}{c_{3}^{2}} \] Verified OK.
Maple solution	\begin{align*} y &= 0 \\ y &= \frac {4}{\left (c_{2} \cos \left (x \right )-c_{1} \sin \left (x \right )\right )^{2}} \\ \end{align*}

Problem 15220

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

Problem 15221

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

Problem 15222

ODE	\[ \boxed {3 y^{\prime \prime }-2 y^{\prime }-8 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {4 x}{3}}+\frac {3 c_{2} {\mathrm e}^{2 x}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{\frac {10 x}{3}}+c_{1} \right ) {\mathrm e}^{-\frac {4 x}{3}} \]

Problem 15223

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

Problem 15224

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

Problem 15225

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 10] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{3 x}+4 \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{3 x}+4 \,{\mathrm e}^{x} \]

Problem 15226

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

Problem 15227

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

Problem 15228

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

Problem 15229

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

Problem 15230

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

Problem 15231

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+10 y^{\prime \prime }+12 y^{\prime }+5 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} x \,{\mathrm e}^{-x}+{\mathrm e}^{\left (-1+2 i\right ) x} c_{3} +{\mathrm e}^{\left (-1-2 i\right ) x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (c_{1} +c_{2} x +c_{3} \sin \left (2 x \right )+c_{4} \cos \left (2 x \right )\right ) \]

Problem 15232

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

Problem 15233

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

Problem 15234

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

Problem 15235

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

Problem 15236

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

Problem 15237

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

Problem 15238

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

Problem 15239

ODE	\[ \boxed {y^{\left (5\right )}=0} \]
program solution	\[ y = c_{5} x^{4}+c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{24} c_{1} x^{4}+\frac {1}{6} c_{2} x^{3}+\frac {1}{2} c_{3} x^{2}+c_{4} x +c_{5} \]

Problem 15240

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

Problem 15241

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

Problem 15242

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }=0} \] 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	\[ y = {\mathrm e}^{-x}+x \] Verified OK.
Maple solution	\[ y = {\mathrm e}^{-x}+x \]

Problem 15243

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

Problem 15244

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }=\left (x -1\right )^{2}} \]
program solution	\[ y = -\frac {x^{3}}{21}+\frac {6 x^{2}}{49}-\frac {c_{1}}{7}-\frac {37 x}{343}+\frac {232}{7203}+c_{2} {\mathrm e}^{7 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {6 x^{2}}{49}-\frac {x^{3}}{21}+\frac {{\mathrm e}^{7 x} c_{1}}{7}-\frac {37 x}{343}+c_{2} \]

Problem 15245

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

Problem 15246

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }={\mathrm e}^{-7 x}} \]
program solution	\[ y = \frac {\left (-x +7 c_{2} \right ) {\mathrm e}^{-7 x}}{7}+\frac {c_{1}}{7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-7 x -7 c_{1} -1\right ) {\mathrm e}^{-7 x}}{49}+c_{2} \]

Problem 15247

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

Problem 15248

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

Problem 15249

ODE	\[ \boxed {4 y^{\prime \prime }-3 y^{\prime }=x \,{\mathrm e}^{\frac {3 x}{4}}} \]
program solution	\[ y = \frac {\left (3 x^{2}+18 c_{2} -8 x \right ) {\mathrm e}^{\frac {3 x}{4}}}{18}-\frac {c_{1}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (9 x^{2}+72 c_{1} -24 x +32\right ) {\mathrm e}^{\frac {3 x}{4}}}{54}+c_{2} \]

Problem 15250

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }={\mathrm e}^{4 x} x} \]
program solution	\[ y = \frac {\left (2 x^{2}+16 c_{2} -x \right ) {\mathrm e}^{4 x}}{16}-\frac {c_{1}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (8 x^{2}+16 c_{1} -4 x +1\right ) {\mathrm e}^{4 x}}{64}+c_{2} \]

Problem 15251

ODE	\[ \boxed {y^{\prime \prime }+25 y=\cos \left (5 x \right )} \]
program solution	\[ y = c_{1} \cos \left (5 x \right )+\frac {c_{2} \sin \left (5 x \right )}{5}+\frac {x \sin \left (5 x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (50 c_{1} +1\right ) \cos \left (5 x \right )}{50}+\frac {\sin \left (5 x \right ) \left (x +10 c_{2} \right )}{10} \]

Problem 15252

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

Problem 15253

ODE	\[ \boxed {y^{\prime \prime }+16 y=\sin \left (4 x +\alpha \right )} \]
program solution	\[ y = c_{1} \cos \left (4 x \right )+\frac {c_{2} \sin \left (4 x \right )}{4}-\frac {x \cos \left (4 x +\alpha \right )}{8}+\frac {\sin \left (4 x +\alpha \right )}{64} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (4 x \right ) c_{2} +\cos \left (4 x \right ) c_{1} -\frac {x \cos \left (4 x +\alpha \right )}{8}+\frac {\sin \left (4 x +\alpha \right )}{64} \]

Problem 15254

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

Problem 15255

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

Problem 15256

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

Problem 15257

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

Problem 15258

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

Problem 15259

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

Problem 15260

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

Problem 15261

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

Problem 15262

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

Problem 15263

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

Problem 15264

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

Problem 15265

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

Problem 15266

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

Problem 15267

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

Problem 15268

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime }={\mathrm e}^{4 x}} \]
program solution	\[ y = {\mathrm e}^{-x} \left (x c_{2} +c_{1} \right )+x c_{4} +c_{3} +\frac {{\mathrm e}^{4 x}}{400} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} \left (x +2\right )+c_{2} \right ) {\mathrm e}^{-x}+c_{3} x +c_{4} +\frac {{\mathrm e}^{4 x}}{400} \]

Problem 15269

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

Problem 15270

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }+y^{\prime \prime }={\mathrm e}^{-x} x} \]
program solution	\[ y = {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+x c_{4} +c_{3} +x^{2} {\mathrm e}^{-x}+\frac {x^{3} {\mathrm e}^{-x}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (24+x^{3}+6 x^{2}+6 \left (3+c_{1} \right ) x +12 c_{1} +6 c_{2} \right ) {\mathrm e}^{-x}}{6}+c_{3} x +c_{4} \]

Problem 15271

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

Problem 15272

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

Problem 15273

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

Problem 15274

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

Problem 15275

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

Problem 15276

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

Problem 15277

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

Problem 15278

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

Problem 15279

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

Problem 15280

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

Problem 15281

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

Problem 15282

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

Problem 15283

ODE	\[ \boxed {5 y^{\prime \prime \prime }-7 y^{\prime \prime }=3} \]
program solution	\[ y = x c_{2} +c_{1} +{\mathrm e}^{\frac {7 x}{5}} c_{3} -\frac {3 x^{2}}{14} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 x^{2}}{14}+\frac {25 \,{\mathrm e}^{\frac {7 x}{5}} c_{1}}{49}+c_{2} x +c_{3} \]

Problem 15284

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }=-6} \]
program solution	\[ y = x^{2} c_{3} +x c_{2} +c_{1} +{\mathrm e}^{6 x} c_{4} +\frac {x^{3}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{6 x} c_{1}}{216}+\frac {x^{3}}{6}+\frac {c_{2} x^{2}}{2}+c_{3} x +c_{4} \]

Problem 15285

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

Problem 15286

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

Problem 15287

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

Problem 15288

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }=8 x} \]
program solution	\[ y = \frac {x^{2}}{2}+\frac {c_{1}}{8}-\frac {x}{8}+\frac {1}{64}+c_{2} {\mathrm e}^{-8 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}}{2}-\frac {{\mathrm e}^{-8 x} c_{1}}{8}-\frac {x}{8}+c_{2} \]

Problem 15289

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

Problem 15290

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

Problem 15291

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

Problem 15292

ODE	\[ \boxed {7 y^{\prime \prime }-y^{\prime }=14 x} \]
program solution	\[ y = -7 x^{2}-c_{1} -98 x -686+c_{2} {\mathrm e}^{\frac {x}{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 7 \,{\mathrm e}^{\frac {x}{7}} c_{1} -7 x^{2}-98 x +c_{2} \]

Problem 15293

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

Problem 15294

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

Problem 15295

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

Problem 15296

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

Problem 15297

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }-2 y=8 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\left (2+\sqrt {6}\right ) x}+\frac {c_{2} \sqrt {6}\, {\mathrm e}^{\left (-2+\sqrt {6}\right ) x}}{12}-\frac {16 \cos \left (2 x \right )}{25}-\frac {12 \sin \left (2 x \right )}{25} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\left (-2+\sqrt {6}\right ) x} c_{2} +{\mathrm e}^{-\left (2+\sqrt {6}\right ) x} c_{1} -\frac {16 \cos \left (2 x \right )}{25}-\frac {12 \sin \left (2 x \right )}{25} \]

Problem 15298

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

Problem 15299

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

Problem 15300

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